Given that we are talking about an animation, on which image is captured frame by frame, we must be careful with some very specific details. Will get more into this details below.
Which tools we need to record a Stop Motion vídeo?
Software
Today, most professional Stop Motion vídeos are recorded using a specialized software, called Dragonframe. Obviously, if you intend to record some videos as a hobby, or if you are only getting started, you may use simpler amature software as well. It's also possible to shoot the sequences without any software at all, and then piece the photos toghether in any standard editing software.
Camera
You may use any digital camera, or even your smartphone.
Tripod
Very important for keeping the camera still, while you shoot dozens, hundreds or even thousands of photos.
What materials and objects do I need?
Here you may use any object at your disposal or, you can make your own characters and and objects with clay, building blocks(like LEGO), paper and other materials. You can use anything to build a scene, the only limit is your imagination.
How to shoot?
1 - Find a good space to the build your scene
Find a space to work, both for building the scene and for recording. Be sure the space is large enough so you can be confortable and that your scenario fits. Be reminded that you will be going around the scenarios various times, and maybe for a few hours. Make sure your camera is well placed, and check the lens are corretcly framing the scene. The camera must be still, and can't be moved during the shooting. If it moves, some photos will be out of frame, and the sequence will look weird.
2 - Ajust the lighting and make sure it's still
Like the tripod, lighting must also be still and unaltered during the shooting. I don't recomend using natural light, because the sun will be changing it's postion and changing the position of the shadows. Do it away from windows, or with well closed curtains. Use continuous artificial lighting.
3 - Be careful while clicking the camera
Configure or camera and keep it on it's designed spot. The ideal way to take the shots is to use a remote control, so as not to touch the camera at any time. This helps in avoiding the risk of ant displacing of the camera.
4 - Slightly move objects within the scene
Move the objects by giving then only small changes in position and, for each movement, take a shot. The ideal way is to move object consitently and slowly. Professional software will help this step with a feature called "onion skinning", that allows to overlay previous and current shot on your computer screen. This way, you may know how much e on which direction you are moving the object.
5 - Edit the sequence
Finished shooting? Now it's the time to load all photos to a video editing software. Load them in sequence, ajust the speed and add sound effects or music as you prefer.
In a more detailed tutorial, I will talk about the ideal frame rate, which software to use and the importance of sound.