That depends significantly on the type of cartoon you are asking about. I'll divide this into three sections to make it easy: Traditional, Computer Generated, and Stop-Motion.
Now the basic principles of all motion pictures are present in the idea of a flipbook, where you have a number of still images that you move through quickly creating the illusion of movement.
Early animated films primarily used cel animation, which involved clear plastic sheets that are painted and then laid over each other. Certain elements could then be animated independently of others. Cartoons like Road Runner or Bambi got a lot of mileage out of the ability to animate background paintings with character cel animation on top. This is the reason why you might notice that an object that is going to move will seem distinct on top of a painted background. So like, when you're looking at a bookshelf in a cartoon you might be able to tell which book will be picked up because it just looks "different."
So the flip-book style is the basis for all modern animation, but new technology makes everything easier. So while some films such as The Thief and the Cobbler actually went all out and hand animated tremendous scenes with hundreds of complicated moving parts and vast armies one could accomplish the same scale for a fraction of the cost:
For example in the stampede sequence in The Lion King or the Hun charge in Mulan they actually only animated a handful of Water Buffalo/Huns and copied them over and over again creating the huge mass of animated movement.
This style of animation is largely how you describe it, with every scene and movement drawn out by hand in some fashion. Now, having a single person hand animate EVERYTHING is really impractical, so you break it down into teams, so you have Storyboard people who plot out the whole movie, then you have animators come in and create the individual scenes. Among animators you get specific people who establish the keyframes or primary gestures, laying out the overall outline of how a motion or action will look. (this is usually the most experienced animator, someone like Glen Keane.) and then a crew of other people will create the "inbetweens" which are the frames that connect the major motions. Then there's cleanup and color and all that jazz, which is handled by more people down the line.
These days most of this work is done digitally with specialized programs such as Toonboom. So Traditional really refers to the frame-by-frame methodology rather than the actual materials used. Few studios still animate with cels (though some Japanese and Korean studios have stuck to the old-school style.)
Unfortunately Traditional animation is very time consuming and expensive so most big companies are moving away from it. Some shows such as Legend of Korra or a lot of French animation such as The Illusionist) still use traditional animation to tremendous effect!
TL:DR Traditional animation is basically a really big and well designed flip book. Each frame is drawn by hand with a bunch of tricks to make it easier or make it so you don't have to re-draw something you already drew. The work is time consuming and falling out of fashion, which kinda sucks, but it'll likely swing back into fashion with new technology or enough built up nostalgia.
CG animated films, such as Jimmy Neutron, How to Train your Dragon, Despicable Me, or Everything Made By Pixar requires related but different skills than traditional animation. Effectively you now have a 3D character model and can animate him like you might an incredibly complicated puppet. However the fundamental skills of a traditional animator are still pretty relevant. Making characters move and interact realistically is a tremendously difficult art form. Again you are creating individual frames and lining them up into a coherent story, however this time you have pre-built models that you can pose rather than having to completely draw the character from scratch every frame. This leads to the ability to move a camera around a character very easily and mimic a lot of techniques used in live-action filmmaking. Things which are much more difficult to do with traditional hand-drawn animation.
In fact, the main hallmark of CG animated films is their ability to render things that are way more difficult to do in hand-drawn animation. Toy Story was pitched with things like shadows cast by Venetian blinds, or the plaid pattern on woody's shirt, the ability to animate with these things was a huge difference in the sorts of things you can work with. Pixar has a tendency to try and push the envelope of complicated stuff. So things like realistic fur in Monster's Inc, translucent human skin in The Incredibles, Water in Finding Nemo, and possibly most impressive of all: Curly Hair for Merida in Brave.
You can think of it not unlike digital stop-motion where your puppets are all digital and way easier to manage.
TL:DR Computer Generated animation uses the same principles of the flip-book but includes some elements of puppeteering and a greater ability to use modern camera techniques.
Possibly the oldest form of animation and I imagine you weren't actually asking about it but I've come this far, so screw it. This style of animation involves using physical models and figures instead of drawn images. You pose a character, take a picture, move the pose a little, and repeat. once you string together the pictures into another flipbook the illusion of movement happens. This gave early animators a lot of the advantages of CG animation before CG animation was even a thing. (that is: a consistent model that can be moved around and stored rather than redrawn every time.) Shows like Gumby, and movies such as The Nightmare before Christmas, or Wallace and Gromit, or Coraline all use Stop-motion animation.
This style of animation is tremendously time consuming but leads to a very unique visual aesthetic that is incredibly hard to mimic. The time and energy that goes into these is a feat in and of itself, and few studios are willing to even try. Laika the company that produced Coraline and Paranorman are still keeping it real though, and they are very careful about mixing in digital effects with their practical models.
TL:DR Stop Motion is made with the same basic flip book fundamentals as traditional and CG animation except instead of drawings they use real physical models.
That's the gist of it, I suppose but someone more qualified than me will probably give you clearer more concise answers.
Sidenote
Flash animation is basically a program designed to make all of the traditional animation stuff way easier. It uses vector based art which is complicated but basically means that everything you draw in it is pretty and smooth. It allows you to draw keyframes and layer scenes and basically everything that traditional animation does, except it happens in the computer and requires way less time to create. You can record and store different animation puppets that can gesture in ways you've pre-programmed into the computer. This style of animation gets a lot of stick because it's becoming more and more prevalent and some think that it leads to poorer animation in general. However it's really just the common uses of the program that are at fault, not the program itself. Flash can be used to create rather beautiful sequences like this.
Rotoscoping One technique which doesn't see much use these days is Rotoscoping, used substantially in films like Peter Pan and Snow White. This involved basically getting a film of actors performing the characters and then drawing cartoons over them frame-by-frame. You'll see this even through movies like The Rescuers. This was considered cutting-edge at the time but fell out of use because the live-action motions ended up less dynamic and interesting than the purely animated characters. (See; the difference between Snow White and the various Dwarves)
