Your question is quite broad but I'll answer it the best I can...
For Laika and Aardman's lip-syncing they use a process called replacement mouths, meaning for each frame a new mouth is created for the said letter the character is pronouncing. This technique sometimes (depending on the character) leaves seams around the characters face so they have to be digitally smoothed out later on in post. Laika does similar only they use replacement faces that have been 3D printed and they are much more complicated.
I would recommend taking a look at this picture which shows an example mouth set. It helped me out when I first started doing lip-syncing.. https://s-media-cache-ak0.pinimg.com/736x/51/6a/55/516a5507347d0004... You can directly replicate those mouth shapes and you'll be doing lip-syncing in no time!
Also here's a link to one of my first lip-syncs after looking at that image.. https://www.youtube.com/watch?v=ClFu5NF1TTE
Here's a tutorial on replacement mouth lip-syncing that could benefit from as well... https://www.youtube.com/watch?v=w6snSRPDJGU