Naturally, a game about taking down giant creatures that can easily be more than 30x your hunter's height is going to have a Crowning Moment of Awesome or two somewhere along the line. As such, Capcom has blessed the series with some amazing musical compositions.
The introduction piece of each game in the series overall follows the same theme, but each have their own unique variations to get you in the mood.
Subzero White Knight (Barioth) plays in the Frozen Tundra, a mix of ethereal chimes and pounding drums. Just watch out of the Saber-toothed Dragon-Cat trying to eviscerate you. You first encounter this theme while fighting Gigginox, but it's equally fitting as Gigginox's fighting style is also wild and chaotic, sometimes involving thrashing about, which is what makes the piano so fitting.
Tremble of the Sea and Land (Lagiacrus) forms the greater part of Tri's intro, and with all the hunting you do in the Deserted Island/Moga Woods over the course of the game, you will be hearing it a lot.
Unique Battle Music is usually reserved for Elder Dragons, but there are a few that have their own themes.
Kushala Daora. A rather unique Monster in that it constantly has a wind "aura" that is difficult to get through without being blown back. Appears in jungles, mountains, and even towns,and capable of conjuring violent storms alongside their appearances. Fortunately, their "auras" can be weakened when poisoned.
A new theme Kushala Daora receives in Monster Hunter Freedom Unite which is even more dramatic.
Lao Shan Lung. In game Lao Shan Lungs are absolutely massive, yet despite their "Elder Dragon" classification, you only ever fight juveniles. You are supposed to defend a town and make your stand at a specially designed fort, and are fully expected to repel it with the fort intact.
Shen Gaoren. A literal Giant Enemy Crab, and is fought in similar conditions as Lao Shan Lung. It usually wears the skull of a Lao like a hermit crab as well, and is comparable in height when it fully stands up on its legs.
Yama Tsukami. The closest the series can get without fully incorporating the Cthulhu deity itself. A floating octopus overgrown with moss and wood that can even summon the swarms of Great Thunderbugs that nest on it. It is acknowledged in game as the oldest living monster.
White Fatalis. A fully mature Fatalis, and requires a hunter to have exceptional timing and skill if they expect to come out on top. One of the few monnsters in the entire series to have a legitimate one hit knock out attack. Can summon red lightning storms as well as periodically shoot an electricity ball from its mouth.
Chameleos. An Elder Dragon capable of staying invisible for an extended period of time unless enraged, then this music starts playing. Chameleos is also notorious for stealing often valuable items from hunters with its sticky tounge.
Nargacuga. The flagship monster of Monster Hunter Freedom Unite. A bat/panther like version of Tigrex, but not as physically strong. Instead, it is insanely fast, and its tail slam can often be a One-Hit KO move for unprepared hunters.
Ukanlos. Calls the icy mountains its home, and can not only burrow through ice and rock quickly thanks to its shovel chin, it even has command over its element. Should a hunter be able to avoid its speed, power, and ice beams, they will find that weapons made from its materials are some of, if not the strongest in the entire series.
Deviljho. Deviljho made its debut in the more recent Monster Hunter Tri, yet has quickly earned infamy as one of the most disastrous things that can occur on a hunt. Can easily wipe out an entire group single handedly, has more HP than the final Elder Dragon in the game it starred in, and his theme even overrides the current battle music. Not to mention it directly rips off The Rite Of Spring in some parts.
Raviente. Currently the largest known monster in the Monster Hunter series, found exclusively in Monster Hunter Frontier. The picture in the video gives an idea of what a hunter should expect in terms of size.
Zinogre. The flagship monster for Monster Hunter Portable 3rd, and it is essentially a lightning wolf dragon. This is also the first track in the series to use guitar, which instead of sounding out of place, adds even more to the overall intensity of the song and the fight against the monster itself.
Moonquake ~ Ceadeus. A heavy tribal drum-beat backing ominous chanting in a made-up language. It underscores the intensity of the fight and the majestic nature of the monster, who lives up to a name that basically evokes "sea god".
From 3 Ultimate/Tri G, we have the Japanese title monster, Brachydios's, theme. The music has the appropriate swelling intensity and epicness of a monster that can kill you by blowing you up.
Also from Monster Hunter 3 Ultimate is the game's final boss, Dire Miralis. The theme really does help convey the sense of power and age that this monster has, and fits the whole "final battle" feel very well.
Tigrex gets a re-orchestrated theme in Monster Hunter 4 that truly captures just how savage and powerful it is.