Humanity is a relatively new species in the galaxy. In spite of this, they are thought to exert undue influence over political decisions. You play Shepard, a Human who becomes the first of our species to rise to the rank of Spectre, a group of elite operatives who answer only to the galaxy's highest political power, the council.
You are tasked with hunting down a rogue Spectre named Saren, who is using the Geth to activate a beacon that will call an ancient race of super-intelligent machines to begin an apocalypse. This apocalypse has occurred before, for aeons.
Along the way, major plot points relevant to ME2 are the works of Cerberus, a rogue human faction notorious for conducting illegal and highly unethical research on alien species in the interest of advancing the Human Race. Shepard encounters one of the Reapers, Sovereign, and defeats it to save the citadel. Depending on your choices, you can either rescue the Council, in which case Humanity will be elevated to council race and you will be a hero to all, or you can leave them for dead, which still secures the council seat for humanity, but the other species do not really appreciate this!
And as for what a Geth is, there is an interesting side quest in ME2 that explores that idea. Essentially there is no singular Geth, they are a series of co-running programs that compound 'intelligence' to become capable of solving increasingly complex problems. Many millions of Geth programs are running on a single platform, platforms being the robotic forms that you spend a large amount of time engaging!