For much of its existence, Mexico has been a Federal Republic in much the same way China is a "People's Republic". Functionally, the government is federalist with a strong central government but states have near complete autonomy and discretionality (this last is a contested sore spot). From bottom to top: Municipal government is handled with mayoral elections and síndicos, or councilmen. This level of government usually shows the a good thermometer for national party preferences. On the state level, each state has a constitution which can be as different from one to another as it pleases the residents, so long as it does not contradict the federal constitution (there's been some brouhaha over the state of Coahuila and Mexico City legalizing gay marriage, with opponents seeking to amend the constitution to force the issue). Each state has a governor and a unicameral legislative. At the federal level, we have a system similar to the US (not coincidentally, post revolutionary governments the world over love to cheat off of the US constitution). On the legislative side there's a bicameral legislature. During the government of the PRI, both were pretty much rubber stamps to presidential initiatives but have gained considerable power over the last two decades after the PRI majority in congress disappeared. Congress is the lower house, and is generally the "rowdy" bunch (with occasional—mind you, occasional—fisticuffs), a bunch of 300 elected congressmen and 200 "plurinominales" or "plural votes", seats accorded to voting preference which the political party staffs as they like. Attempts to winnow this ungainly number have been... difficult. The Senate or upper house is much smaller at 128 and generally more serious (which isn't saying much, it's distressingly common for legislators in either house to seize the tribune). For much of its history all the real power rested with the President, a six year dictator with supreme power over the PRI (the main political party from 1920-2000) and the government. Nowadays, the president still has a great deal of power... but it's the legislative who's boss. The thing about the six year dictators is that most of their power was never codified, and in fact it's the legislative that holds a lot of the weight. This is something the legislative is just figuring out in its adolescence, and the result looks rather like the US in the period between independence and the Civil War (you know, when brawls on the House floor broke out every now and again, and a Senator was beaten within an inch of his life, but the legislature controlled all).
PartiesMexico is a multi-party democracy, and since 1997 (when the PRI weakened) no party has held a majority in either house, so most legislation has to go through on a case-by-case basis. So far, all presidents were either from the PRI or most recently from the PAN (latest two). The parties are by 2006 voting preference.
The PAN - A conservative party with Christian roots, it started sometime in the 40's as an elitist party formed by doctors and lawyers and other 'moral individuals'. Nowadays they're much more mainstream and Center-Right in affiliation. Those who support them cite their Catholic virtues and inspiration as well as their straight, upstanding morals. Those who don't cite this is just a hypocritical facade to secretly do nasty businesses in the background, as well as a perceived lack of concern for impoverished citizens. As of 2009, the PAN lost one of its largest historical supporting states after the governor of Jalisco publicly dropped a Cluster F-Bomb at those who criticized his nine million dollar "macroalms" for the construction of a Catholic temple, which was widely vilified as a violation of the secular state principle by those who are on the left, and a violation of the party's upstanding morals by those who are on the right.
The PRD - A moderate left to extreme left party. It split from the PRI over their inability to provide actual democracy and the then recent shift to the right in the PRI. Cuauhtemoc Cardenas, the PRD's founder, left because the tradition that the outgoing president picked his successor both cost him a shot at the presidency and disgusted him. Those cynical enough will tell you that the PRD was originated by PRI members discontent with their meager lot in the old party's distribution of embezzled money and influences. Nowadays they're a hodgepodge of "tribes" ranging all over the left. This internal division has grown particularly nasty, since the two largest tribes basically accused each other of stealing the internal election. As of 2009, they are a single party in name only. The defeated presidential candidate Lopez Obrador has firm control over half the party, despite having alienated most of the middle class, causing a reduction in voting preferences to 14% of the electorate. He's opposed by the "Chuchos", and together they have divided the party with no clear reconciliation in sight. Many political analysts doubt they'll be able to field a credible presidential campaign come 2012Update: Lopez Obrador was candidate for the PRD in 2012 after all.
The PRI...probably the only political party in existence without a clear platform.note When it was first formed in the early twenties, it was quite communist — they let Leon Trotsky take shelter in Mexico for a reason. Then, for decades, it was undefined and anti-communist, but other than that it accepted people from any political viewpoint as a means to appeal to everyone. After 1994 with the economic crash, the party has been floundering without strong leadership and internal divisions making unified candidacies difficult. In 2006 they had a terribly poor showing because the Party president forced himself as candidate and earned the ire of all the other pre-candidates. Their fortunes are looking up, as the decomposition of the PRD means they can act as the "majority maker" for the PAN. After the 2009 midterms, they have gained a plurality in the lower house, made into ade facto majority via their alliance with the PVEM. Along with control of more than half of the country's governorships, they are once again serious prospects for the presidency. Just to repeat that: the political party that held a 70 year long dictatorship over Mexico may be willingly voted back in. Though explanations for this startling comeback are many, it's mainly attributed to the 2007 electoral reform, anger at the world economic crisis, a very poor performance by the PAN originated presidents, and several PRI state governors using decades old and brand new vote buying techniques that the authorities can't or won't stop. What they'll do with their newfound power is anyone's guess, most think they will go full tilt into deep reforms to give their 2012 candidate better odds of inheriting a solvent country... others, that they will only change things just enough to keep the state afloat without letting PAN president Calderon reap much benefit, while tilting the laws in their favor. There's a sundry list of smaller parties, including the PVEM (Green Party, which is the ecologist part of Mexico but is ironically promoting the legalization of death penalty, and has been criticized as being the front of a family mafia), PT (Workers party), etc. None have yet gotten a governorship, but a lot of a dozen or so seats in the legislature.
Issues:Currently the most pressing issue nation wide is insecurity and the violence brought in by the infighting between drug cartels and federal government efforts to curb these cartels. For decades the police forces have been neglected at local and state level resulting in police forces that are substandard even by Third World levels. Municipal and state police are underpaid and poorly armed, being no match for the drug cartels who supply themselves via illegal gun smuggling mostly through the United States. These leaves the federal government forces (including the Army) virtually alone to fight the huge mafias that developed over the country for decades sometimes protected by local state governments and bureaucracies. Traditionally, the biggest issue has been extreme poverty and corruption. Income-wise, Mexico is among the most unequal nations in the world and decades of government intervention to supposedly help solve it have only resulted in corruption. Energy has always been the most sensitive (if most of the time dormant) issue; in Mexico oil production, from exploration, to refineries is by law exclusive state property and seen by many people as a bona fide national symbol not to be let into private hands, let alone foreign. This has stalled much needed reform in the energy sector and squandered any economic advantage Mexico might enjoy as an oil producer. Oil production is expected to last around 20 years more (as of 2012) before it exhausts. With the upcoming July 2012 presidential elections, the strong favourite to win the presidency is the PRI. The electoral reform of 2007-2008 has made the process more expensive and alienated voters and the media monopolies by flooding the airwaves with advertisements the government forced the media to show. The petroleum reform has met with very little success or had a discernible impact, with projected oil production to drop, and gasoline, gas, and electricity prices to rise. Since the PRI-PVEM alliance has a majority in Congress, odds are these systemic deficiencies will not be adequately addressed (Both are pretty much against energy sector reform, and being the big winners under there current electoral system, no incentives there). It remains to be seen if they create any worthwhile reforms in the Energy, Health, Education, Government and Labor sectors that don't cater to the syndicates/oligpolies. The Zapatistas (EZLN) are an issue that shrinks day to day. Taking their name from indigenous revolutionary leader Emiliano Zapata, the Zapatistas are a group of indigenous people in the deep southern state of Chiapas who rebelled prior to 1994 in order to gain international recognition for the plight of indigenous peoples. Though not a political party, the Zapatistas have been accorded governmental autonomy for the municipalities they control, making their tribal council rule a political force at least locally. The cliff notes version of their plight is this: for much of it's pre- and modern history, Mexico has never known what to do with "the indigenous problem": remote indigenous populations with high levels of poverty, illiteracy and no infrastructure, leading to systematic neglect and abuse. They rebelled (with actual arms and a few battles against anti-Zapatista paramilitary groups and the army) and a few infuriatingly obscure massacres happened to/by them (that's the infuriating part, many aren't sure if they are La Résistance and not a wannabe The Revolution Will Not Be Vilified). However, after the initial gains made negotiating with the government, they have languished and dropped off the radar except as political/military/cultural curiosities, and the occasional "Zapatista tour" or cryptic pronouncement by their leader Subcommander Marcos. News reports suggest they are becoming an entrenched, corrupt political elite among the ruled municipalities, while ironically, the government's earnestly pumping effort and money into improving the situation for all non-Zapatista municipalities has given them a marked improvement in standard of living. Even if it's only to show up the Zapatistas, their getting better results is telling.