A.k.a. the Cleopatra you're thinking of.
Cleopatra VII Philopator
was the last Pharaoh of Ancient Egypt
, at different points ruling jointly with her father, brother, other brother, and son, though with the latter three, she was clearly in charge and only using them to bolster her credibility as a female ruler. The Ptolemaic dynasty Cleopatra descended from was actually Greek; after the death of Alexander the Great
, his empire was divided up by three generals, and Ptolemy I Soter got Egypt. Cleopatra was actually the first of her dynasty to bother learning Egyptian, and presented herself to her kingdom as a reincarnation of the goddess Isis
The identity of Cleopatra's mother is uncertain. Born the third child of Pharaoh Ptolemy XII Auletes, Auletes lost his grip on his kingdom due to corruption and the loss of of Cyprus and Cyrenaica. In a desperate bid to regain control, he fled and begged Rome for money and troops to help him regain his throne. Cleopatra's two older sisters, Cleopatra VI Tryphaena and Berenice IV, seized power at this time — first Cleopatra VI, then Berenice upon her mysterious death.
Whether or not Cleopatra VII accompanied her father to Rome or remained in Egypt is debatable; she isn't really a concern in the contemporary records of either place. The reason this is even a question is because some accounts describe her as meeting Marc Antony around this age, while others assert she met him as an adult.
What is certain is that Auletes eventually was able to secure the troops and money, and that Berenice was imprisoned and executed for her disloyalty. Cleopatra was now fourteen, the eldest of her remaining siblings, and thus the one with the best chance of keeping a hold on the throne should their father die. He elevated her to joint ruler at this point, though it's unlikely she had much power.
In his will, four years later, he decreed that 18-year-old Cleopatra would rule jointly with her ten-year-old brother, Ptolemy XIII. She married him according to Egyptian custom, which the Ptolemies adopted for whatever reason, but refused to share power with him. Eventually, she progressed to leaving his name off official documents and his face off the coinage. This was just another complication on top of widespread famine and the Nile not flooding enough to irrigate crops.
Around this time, Cleopatra ran foul of the Gabiniani, troops left behind when her father was restored to power. They killed the sons of the Roman governor of Syria, she handed them over in chains, and they didn't forget that. She was exiled when they joined with a cabal of courtiers and had Ptolemy placed on the throne as sole ruler.
Aged thirteen, Ptolemy was eager to establish himself. The Roman civil wars between Pompey and Gaius Julius Caesar
were raging about this time, and Pompey took the gamble of begging Ptolemy for succor. Ptolemy was waiting seated on a throne in Alexandria harbor when Pompey disembarked, and ordered him executed on the spot, while his wife and children watched from the ship. Ptolemy hoped to gain credit with Caesar by killing his enemy for him, and allying himself with Rome— which, you remember, Egypt borrowed a lot of money and troops from in the previous reign.
When Caesar arrived in Alexandria, he was presented with Pompey's severed head.
He was pissed.
Whatever Caesar's political battles with Pompey, Pompey was a Roman Consul and his former son-in-law, his only legitimate child having been Pompey's wife. Believing that Ptolemy had no right, Caesar promptly took control of Egypt and declared himself the mediator between Ptolemy and Cleopatra. Cleopatra saw her chance and, legend has it, had herself smuggled to Caesar inside a rolled-up carpet. They had an affair, with Cleopatra giving birth to his son, Caesarion, or "Little Caesar." End result was that Caesar decided not to annex Egypt, and settled that Cleopatra and her brother were to be reconciled. Cleopatra would have preferred for Caesar to name their son his heir (which was illegal under Roman law, as Caesarion was not a Roman citizen) and to rule jointly with him after Caesar left, but Caesar refused.
This led to a civil war in Alexandria, and, possibly, the burning of the famous Alexandria library. Caesar eventually got things under control, but Ptolemy drowned when his armor weighed him down too much. Cleopatra was promptly married to their younger brother, and he
became her co-ruler. Her younger sister, Arsinoe, was taken to Rome by Caesar and executed.
- Adaptational Attractiveness: There's no doubting that she was a phenomenally sexy woman, but she definitely did not look a thing like Elizabeth Taylor.
- Asskicking Equals Authority: Had to win several battles against her brothers to become sole ruler of Egypt. Eventually only supplanted by the first Roman Emperor, so also undone by this trope.
- Badass: Gosh, where to start?
- Beam Me Up, Scotty!: Modern perception of Cleopatra has been distorted quite a lot by her most famous film - the real life Cleopatra would not have worn the bizarre headdresses and hair accessories that Elizabeth Taylor wore (she preferred the simple diadem worn by Hellenistic monarchs), and did not have bobbed hair.
- Better to Die Than Be Killed: Famously offed herself with a bite from a snake. The alternative would likely have been to be paraded through the streets of Rome in chains before being killed in some creative manner by Augustus' lackeys.
- Betty and Veronica: Most certainly the Veronica, where the more decorous Roman Octavia was the Betty and Marc Anthony the Archie.
- Big Entrance: Since she was at war with her brother/ husband, she was forbidden from entry into Alexandria when Julius Caesar was visiting. Instead, she was smuggled into the city in a rug, which was unfurled in front of Caesar to reveal the Queen hidden inside. The effect on the notoriously amorous Caesar of being surprised by a beautiful 21-year-old must have been quite astounding.
- Brother-Sister Incest: The Ptolemies practised inbreeding as a tradition inherited from the original Pharaohs of Egypt. Cleopatra herself married two of her brothers, although given her second husband-brother was 14 when he was killed, this marriage was likely never consummated.
- Cleopatra Nose: Trope Namer, of course. A trait she inherited from her Ptolemaic ancestors, she tended to exaggerate it in portraiture to stress her family ties.
- Driven to Suicide: Took her own life after seeing Antony die and Egypt conquered.
- Ethical Slut: Cleopatra was an adulterer and broke up Marc Antony's respectable Roman marriage, but she was a serial monogamist who seems to have genuinely loved Antony.
- Face Death with Dignity: Her response to seeing the snake used to kill herself was "Oh, so this was here."
- Gag Nose: This is usually how her nose is depicted in comic strips and cartoons. In the Asterix franchise it is drawn as a pointy triangle.
- Going Native: Contrary to popular belief, she was not herself Egyptian, but a Hellenistic Greek monarch, ruling in Egypt. Being on the tail-end of centuries of culture-mixing between the indigenous Egyptians and Greek colonists, Cleopatra's court was a mix of Hellenistic and Egyptian tradition, and she was the first Ptolemy to speak Egyptian in public and see much of the Egyptian countryside.
- Klingon Promotion: She got to the top as the end result of decades of civil war in Egypt.
- Lady Looks Like a Dude: There are two kinds of ancient portrait of Cleopatra - ones in which she looks like the epitome of Greek femininity, and the ones where she looks like a cross-dresser. The latter was to stress her links to her predecessors as king, imitate images of Roman consuls, and to show herself as having the heart and stomach of a king.
- Last of Her Kind: The last ruler of the Ptolemaic line in Egypt, and the last Pharaoh.
- Memetic Sex Goddess: Manage to attract the most powerful men in the Roman Republic, and even had one of them betray his nation and kickstart a civil war to be with her.
- Pop-Cultural Osmosis: Together with Nefertete and Hatsjepsut the most famous female pharao in popular culture, though she even eclipses them.
- Race Lift: The real Cleo looked very Mediterranean, but she is popularly imagined either with Raven Hair, Ivory Skin, or as a black woman, depending on the artist.
- Royals Who Actually Do Something: She didn't leave her mark on history by sitting around eating figs, now did she?
- Inspired a catchy jazz song, Le nez de Cléopâtre (1938) (Cleopatra's nose) by Ray Ventura. 
- The 1963 film Cleopatra with Elizabeth Taylor is proably how she lives on in the imagination of most people.
- The Astérix album Asterix and Cleopatra introduced her as a recurring character in the Asterix franchise.
- Also battled against Marilyn Monroe in Epic Rap Battles of History.
- Plays an important role in the second series of Rome where she has sex with Titus Pullo.
- Star-Crossed Lovers: The love-affair between Cleopatra and Marc Antony has become one of the most famous historical romances.
- Vindicated by History: More of a household name than many, if not any, of her contemporaries.