Wow. That is a massive set of questions. Here are the short answers, please note that I am skipping lots of important specific information.

Dailies are quests you can repeat everyday. There are dungeon dailies (through the undaunted and random dungeon finder), there are crafting dailies and dailies in the base game for guilds; these generally increase your skill line for the respective guild or activity, and they are a source of gold, gear and materials, and crafting dailies can give master writs for housing. Each dlc zone also has a selection of dailies that usually offer rewards from there gear sets and rare motifs, Wrothgar is popular at the moment because you get double rewards during the anniversary event.

Here is an incredible resource on all the daily quests. http://benevolentbowd.ca/things-to-do-in-tamriel/

Trials are 12 man dungeons that you can only do with a group of players you choose to group with. They can be done on normal or veteran mode but they are harder than most other content in the game.

There is also the maelstrom arena which is a single player trial and the Dragonstar arena which is a 4 man trial. Completing veteran trials rewards some of the best weapons and jewellery in the game.

Veteran is the higher level of difficulty for trials and dungeons on. It is endgame pve. Normal mode is the easier mode, and you can't do anything veteran until you get to champion level.

champion is the level you get after level 50. After level 50 you earn champion points which you use to slightly increase different stats. At 160 champion points you can wear the highest level gear. At present the maximum number of usable champion points are 690.