In the most current versions of Minecraft, there are three main game modes. Survival mode, Creative mode, and Adventure mode. In survival mode, you have to mine resources and you can get hurt. This game mode is the most common for actual game play and is the default game mode. In creative mode, you can't get hurt, you can fly around, and you have an infinite amount of resources and blocks at your disposal. This game mode is mostly used for building anything you want. Adventure mode is about the same as survival, except, in adventure mode, you can't break any blocks. This game mode is mostly used in you are playing on an adventure type of map, and the creator doesn't want you to break anything in the world.
Minecraft has three types of mobs, passive mobs, hostile mobs, and neutral mobs. Passive mobs won't ever hurt you, and some examples are animals such as pigs, cows, and chickens, and NPC's (Non-Playable Characters). It is a good idea to stay away from hostile mobs, because they will attack you no matter what. Examples of hostile mobs include monsters such as zombies, skeletons, creepers, etc. Neutral mobs are mobs that are passive, but become hostile if you attack them. Neutral mobs are wolves, endermen, iron golems, and zombie pigmen.
In Minecraft, a biome is a climate zone used in the game to set what kind of surface the ground has (sand? grass? stone?), whether it should rain or snow, what trees grow there, and sometimes also what kind of animals that are allowed to spawn there. Some snowy biomes are ice plains and cold taiga. Some hot biomes include the desert, savanna, mesa, and hell(nether) biomes.
Link to Minecraft Wiki