Action cards are cards that do their effect and then disappear. They do not last long and do not generate studs at all, unlike most fighters, and are thus cards that should be used sparingly; Too many actions can wither your early game and opening hand. It is likely, however, that they will have a big impact on the game's state, whether the action cast was a killspell to deal with a threat or a boardwipe to punish overextending.
Most control decks will have ways to directly counter actions or punish for using them, such as Eisenhower Reunited or Xenotrent. Players typically make use of a few actions to support their deck, but Groups like Lunar, Lycanthropes, and Apocrypha revolve almost entirely around actions.
Terrain cards are cards that affect you during either each or your turn. These provide passive effects over the course of the game and are best played early. Most terrains do not naturally generate studs for you, and thus, having too many will wither your early game and opening hand. However, it is almost guaranteed that your terrain will last the entire game, as there are very few ways to directly destroy one, such as Melon_Desu.
There can only be one terrain in play that you control at any one time; Playing another terrain replaces the one currently in play. Most of the time, players generally don't use terrain cards, but Groups and decks like Caverns, Terrarians and World of Wood love them.