Classic Single Line Slots
Basically what it says on the tin. The original three reel with one payline across the center. Became famous in the casinos over half a century ago
Classic Multi Line Slots
The same as above but will have up to 5 paylines across the 3 reels. More chance of winning but of course you have to place a bet on every line.
Standard Video Slots
These will be 3 or 5 reel slot machine games but instead of having the fixed reel effect of a classic slot machine the reels will be "virtual" and are in fact a video screen made to look like a spinning reel.
Bonus Game Video Slots
The same as above. A 5 reel game with multiple paylines. A certain combination of symbols on a payline will start a bonus game round to win you extra cash.
Multi Spin Video Slots
1 single payline but you have a chance to hold symbols and spin again. The symbol can be held across several reels for a bigger payout. Similar to multi hand video poker.
Progressive Slots
Any slot game played at a casino can have a progressive element built into it. What this means that every bet you make a small percentage is placed into an accumulating progressive jackpot fund. This fund keep growing until someone wins the jackpot.