Omaha is a variant of Texas Hold'em Poker. Omaha is considered, by some, to be the hardest game of Poker to master. Of all of the different games of Poker, Omaha is for many the hardest to learn to play and the hardest to bluff in. It is played most often at fixed limits, like Texas Hold'em and pot limit.
A run is a sequence of games, especially wins. A deep run into a tournament is a long sequence of games in the tournament, which takes one "deep" into, meaning a long way into, the tournament. This is most often a long run of wins in the tournament, as very often one loss is enough for elimination.