A move (also known as an attack or technique) is what Pokémon have primarily used in battle ever since the Generation I games. In battle, a Pokémon uses one move each turn. Some moves (mainly moves learned by Hidden Machines, though not always) can be used outside of battle as well, usually for the purpose of removing obstacles or exploring new areas.
Moves have always been learned by leveling up, by use of Technical and Hidden Machines, and some Pokémon can learn moves only at certain evolutionary levels. Since Generation II, they can also be learned through breeding and from move tutors. Smeargle can also use a move called Sketch to learn a move in battle.
A Pokémon can only know four moves at a time. The moves which each Pokémon can learn is dependent on the Pokémon's species. In battle, moves can cause damage, affect a Pokémon's stats, heal the user, make the opponent flee, etc.
Several Pokémon learn moves by level up at levels lower than that at which they can be obtained, either by capture (the case with many legendary Pokémon) or by evolution. But here at Blood Sport, more things are taken into consideration with battling, such as body mass, type, exc exc. (A Venusar using bodyslam against a Chimchar is gonna mess up the monkey)
As far as "new moves", remember that this is only like the games in the sense that you will have 4 moves, the same Pokemon, and similar items. Where you can't create a new attack, you can be creative with attack combinations to create new effects. An attack that is something of a straight shot like water gun could be amplified by movement (i.e spinning, some sort of launching attack). So where you are stuck with your Pokemon's move set, a crafty assortment of moves could prove more victorious than straight up strength (thanks to Jade for that tidbit)
For a list of all pokemon moves, please look -here-
(Thanks to Bulbapedia for the information)