We've never heard the term "client," but it's obvious that the discussion was about playing with "backers." A backer (or "investor") is someone who finances a skilled player for a cut of the winnings. Backers will get involved in any game or gambling situation where skilled players have a mathematical expectation of coming out ahead over time. A player has to be known to be good enough to be backed, but when he is, it's a good way to build his own gambling stake, since the backer typically absorbs all losses if a profit isn't realized.
Backers run the gamut from former players who've moved on to other things and want to continue earning from their field of expertise, to players who simply don't possess the skills to win, but have a lot of money and see this as a legitimate form of investing. It's very common in poker, where new players who exhibit skill seek to be "staked," either by other established poker players or by one of the big online poker sites. It's very common in poker tournaments, and in the late stages of a big events you'll hear people talking about how many "horses" they have left, meaning players they're staking that are in position to earn them money.
Using your video poker example, here's how it works. Your backer might give you $10,000 to play an opportunity where a progressive meter has gone positive. That $10,000 is called the "bank" and it might, depending on the deal, also be used to pay expenses like food or travel if the game is in another city.
Once the play is over (say the jackpot gets hit), the bank is "broken." If someone else hit the progressive, then you probably lost money and whatever remains is returned to the backer at that point. However, a "make-up" is usually in effect. That means that if you make a similar deal in the future and it's successful, the player doesn't get a share of the win until the prior loss is recouped.
If it's you that hits the progressive and there's a profit, it's shared according to the negotiated terms. The split percentages are structured in many different ways depending on the amount of the investment, time spent by the player, etc. (and it can get especially complicated in video poker because of the tax ramifications for hitting big jackpots), but it's often as simple as cutting up the winnings 50/50.
Of course, trust is a paramount component in this type of arrangement. Since the backer is rarely in a position to monitor results, he's subject to false reporting and many a player-backer deal has dissolved over pure suspicion (there's an interesting story about this in Bob Dancer's Million Dollar Video Poker). Because of this, it's difficult to get someone to back you that you don't already know well.
