We talk about an asymmetric game when players play with each other or have different goals. We can highlight games partly asymmetric or completely asymmetrical. The easiest way is to describe the issue on the examples. We usually consider this issue with respect to PVP games, not game types player vs game.
Starcraft is an example of a partially asymmetric game. All players use the same resources (crystals and gas) and have the same targets (usually eliminating opponents). All players are also subject to certain general rules of the game mechanics such as fog of war, respect obstructions, produce units and buildings etc, etc. The asymmetric element are races (Zerg, Protoss and Terran), which have different units and styles of play.
For example, the game completely asymmetrical let us use the Evolve. Here, the Łowczych group is put in front of one strong monster. The hunters are separate characters much weaker than the monster, they use completely different mechanics of the game and weapons. Players ' goals are also different. Hunters must beat poczwarę before he can achieve the full force, and the role of the monster is to avoid chasing him and hunt small creatures to grow in strength.
I also want to encourage you to study classic symmetrical games such as checkers or chess and asymmetric ones such as Fox and Gąski.
Another interesting type of asymmetric games are games in which players start with the same position, but their possibilities begin to differ on the icy of their choices. A great example is card games where players build their blankets from among the cards that are available to all, and then fight against other people (Heartstone).