I've not played Dominion, so I can't speak for that, but I am familiar with Ascension - it was, in fact, my favourite out of the card games we owned…until I got hold of Sentinels.
In Ascension, everyone starts out with a deck of ten cards, consisting of eight Apprentices and two Militia. You have your deck in front of you, with space for a discard pile, and draw a hand of five cards. Obviously, your starting hand will consist of either five Apprentices, four Apprentices and one Militia, or three Apprentices and both Militia. The only Apprentice and Militia cards in the game are those in the players' starting decks - they can't enter the game by any other means so you just keep the rest off to one side/back in the box.
The game has a board, on which are laid all the other cards. There are three card types laid face-up to one side of the board. These are Mystics, Heavy Infantry, and a Cultist (of which there is only one in the game). The rest of the cards (the "deck proper") are placed in a stack at the other end of the board, and the top six are dealt, face-up, along the middle row of the board for the players to see. You decide who goes first and the game starts.
On your turn, you can buy cards or defeat monsters. Your resoureces for buying cards are called Runes. An Apprentice produces a single Rune. So if your starting hand consists of five Apprentices, you can buy five Runes' worth of stuff. The Mystics cost three Runes to buy, but generate two Runes each (so you can use three Apprentices to buy one Mystic). So if you buy Mystics, you can use them later to buy more expensive stuff.
On the other hand, defeating monsters requires Power. A Militia has one Power, so it can only defeat a creature requiring one Power to defeat (numbers are shown on the cards). Heavy Infantry cards (the stack next to Mystics on the board) cost two Runes to buy and generate two Power, so you can use them to defeat more powerful opponents. The Cultist only requires one Power to defeat so you can always just kill it a few times if you have attacking cards in your hand and nothing to kill - Cultists are considered to be an infinite supply so you just leave the one card on the board for the entire game.
Anyway, you use your starting hand to buy/slay cards from the centre row, or buy Mystics/Heavy Infantry or slay the Cultist. When you buy a card, it goes into your discard pile. Your turn ends when you've used (or discarded) all your cards - you don't have to play a card if you don't want to, and indeed can just discard your entire hand without doing anything if there's nothing on the board that you want.
Next round, you draw the next five cards from your deck, the other half of your starting ten, and buy/kill more stuff. Next round, you have no more card in your deck so you shuffle the discard pile and lay that down as you would in Sentinels, now drawing the top five cards as your new hand (which will of course hopefully contain at least some of the stuff you bought during the first two turns). These might consist of Heroes (one-shot cards who do one thing and are then discarded) or Constructs (equipment cards which stay in play once played, and whose abilities can be used as described on the card, usually once per turn). Both types of card can produce more Runes and/or Power, or may have some other effect (such as letting you draw more cards from your deck or insta-killing a monster with a certain number of hitpoints).
Whenever a creature is killed, the person who killed it gets some Honour (in the form of a little plastic token) as a reward. When the Honour total (in tokens) reaches a certain total depending on the number of players (60 for two players, I think 75 for three and 90 for four), the game ends at the end of that round. Everyone then totals up their Honour (bought cards also have an Honour total, which is only counted at the game's end) and the player with the highest Honour total wins :).
Now someone needs to outline the rules of Dominion so we can see what the comparison is :D.