Sometimes you will see passive as weak and aggressive as strong, so be aware that the terms weak/passive and strong/aggressive are synonymous.
Tight-Passive: This is the player that plays least in the game of poker. These are people that will play nothing but the best of starting hands and even when they do play hands they are likely to only call or raise the minimum with those starting hands. The only time a tight-passive player will raise big is when they are holding the absolute nuts. Tight-passive players are also sometimes referred to as rocks and while rocks can be marginally profitable players at the lower limits, they will get flayed alive at the higher ones.
Tight-Aggressive: This is the style of poker that is ideal for the vast majority of people. The vast majority of professionals play some sort of variation on a tight-aggressive game. Tight-aggressive players only play the best of the starting hands just as in the previous example, but the difference is that they bet big when they have the best hands. They bet with the best hands, call with good draws and throw everything else away. Tight-aggressive players will rarely bluff, although you might catch them in a calculated semi-bluff from time to time; especially on the penultimate street.
Loose-Aggressive: A loose-aggressive player is the only other type of player that can make money at poker aside from the tight-aggressive player. When playing poker, you need to play aggressively regardless of what your starting hand selection is; playing weak in any form is not going to get the job done. Loose-aggressive players are players that see a lot of hands, but bet a lot and raise a lot in order to throw their opponents off stronger hands. Loose-aggressive players are often big bluffers and depending on how effective they are at bluffing, they can make the majority of their money off of that one move. While loose-aggressive professionals are not as numerous as tight-aggressive professionals, a lot of them do exist. Gus Hansen and Sammy Farha are two good examples of loose-aggressive professional players.
Loose-Passive: The absolute worst combination imaginable is this one. This is a player that sees a lot of flops, but at the same time does not play strong at all. Tight-passive players will at least be able to catch good hands that hold up because they are only playing the best starting hands, but loose-passive players will lose a lot more money because they will be calling with second and third best hands on a frequent basis. Make sure that you never play this playing style, because it is almost guaranteed to end up badly for you unless you catch an extraordinary run of luck.