There is no one-true way to shoot in football, the only correct way is the way that works (i.e. the way you can score the most with).
That being said shots can be classified with the part of the foot that comes in contact with the ball, as SahuKahn has mentioned in his answer. I'll give more details:
Inner side of the boot: the most classic shot with the inner side of your boot. Here's a video explaining this type of shot. Pros: typically best accuracy, most control on the ball thus best chance to "place" the ball. This type of shot is also easier to fake and continue with a dribble.
"Knuckle" shot: name is pretty self explanatory really. Here's a long video showing examples and some explanation as to how to shoot like that. The advantage is that the ball path is really difficult for the goalkeeper to judge, and the drawback is that it's hard to "get it right". Usually you need a good and controlled run up to the ball, thus this type of shot harder to pull off in the heat of the game.
I have personally tried it out and I can see a definite improvement of my shot even tho I am not quite there yet..
Lace shot: This would be the footballs equivalent of a "smash". I would call it a typical, old-school English shot with Steven Gerrard and Frank Lampard being two kings of this type of shot. Here's a video comparing those two players (the first 2 mins is relatively irrelevant fan stuff). Pros; full power on the shot, practically the entire motion of the leg is transferred to the ball. Cons; hard to control the accuracy, also ball path is usually pretty much straight
Outside shot: This is actually pretty much the opposite of #1. Outside of the boot, very little control (usually) and you get the reverse curve on the ball. Good example of such shot would be to look at Quaresma, here's a small collection of goals he scored.
This type of shot is unpredictable and hard to save for the goalie and it typically generates a counter-intuitive path. However it's also extremely hard to get it effective and outside of the boot is typically not the best part of the foot for control. For instance I was told on multiple occasions to keep my outside-foot dribbles/shots/passes to an absolute minimum.
Chip: You use the front part of your boot and go under the ball, to lift the ball quickly. This type of shot is used to "chip" the ball over the goalkeeper, either because the goalkeeper is far off the goal (in comparison to where the ball is) or that goalkeeper is charging to sweep the ball. Here are some examples.
Besides these you can of course shoot with your toes, or heel etc, but that's not that much different than shooting with your knee or shin; the ball hits you somewhere and if you are lucky and get a "clean" hit then the ball bounces back towards the goal.
I would not recommend relying on a toe-shot (even though we have seen several scored this world cup like Oscar's goal vs Croatia) as it's weak and typically very straight ball path.
As for how to improve: practice makes perfect. Look for drills, take shots against a wall, against targets, ask tips and pointers from friends. I typically play with the ball before and after the games, and aim at the goal, the posts, the crossbar, any signs or lampposts, even friends. Even that counts as practice... The more shots you take, the better your foot and your brain understands the ball paths, and amount of power you need to put in to get the shot right.