There are many things that can cause problems with a bike chain.
Bike chains take as much wear and tear as any other part of a bike. Chains need to be taken care of by lubing and cleaning them between rides. Correct instillation is also essential. It is recommended that you replace a chain every 1,200 to 1,500 miles to keep chain snap and other chain problems from occurring.
Chain Skip/Slip
There are three things directly related to your chain that may cause it to skip or slip. If the chain is the improper length (too short or long) or not installed properly, your bike will not be able to shift properly. This can cause your chain to slip/skip. The way to remedy this is to add or remove links in your chain with a chain tool. If your chain is heavily worn, it should be replaced.
If your chain is dirty, not lubed or rusty, it might be causing the skip. Chains need to be lubed after each cleaning and every few rides in between. If dirt and other grime has built up on your chain, the chain can slip. Make sure to clean your chain when you clean your bike, especially after muddy rides.
If you have any links that are too tight or lose, your chain may be kinking. Loosen or tighten any needed links to see if that stops the kinking.
Chain Stretch
As your chain flexes and straightens while it moves on and off the sprockets, the metal wears away over time near the location where the rivets rotate. This will cause the chain to stretch as you log miles on the bike. This means your chain needs to be replaced if you want it to work properly.
Damaged or Stiff Links
Links can get damaged from a crash, travel and other causes. Bent links need to be replaced. If you have leftover chain links from your original chain install, you can use them to replace the bent links. If you don't have any extra links, you will need to buy a new chain.
Stiff links can be fixed by hand or by using common household pliers. To lessen stiffness, take the chain in your hand or hold with the pliers and move it back and forth to loosen any grime built up on the chain. Finish by lubing the chain.
Chain Snap
Chain snap usually happens for one of two reasons--a bad install or wear and tear. If the chain isn't put on correctly, any force placed on it can cause it to snap. If your chain is old or damaged, it can also snap apart. It is recommended that you replace the chain if it snaps instead of trying to just replace links.
Wrong Chain
When installing your chain, you need to make sure it is the right speed and length you need. Specific chains are made for nine, 10 and in rare cases the 11 speed bike, because of the spacing of the cassette. Also if the chain isn't the correct length, the tension can be off and can cause the chain to skip while shifting.