Whether you'll be able to cash a torn check depends on a number of factors, notably how severely the check is torn and the policy of the bank you're using. Torn checks rarely can be deposited at ATMs, so you'll have to go to a teller to present it for payment. The teller and perhaps a manager will look at the condition of the check and decide how to proceed.
Go to the Issuing Bank
You may have better luck at the bank that issued the check than the one at which you keep your own funds, since it will be able to verify the check's authenticity more quickly. That bank may take additional steps to confirm the check is good, such as calling the account holder. If you take it to your own bank, the bank may place an extended hold on the funds until it verifies that the amount is good, which may mean you'll have to deposit the check rather than cash it.
Torn or Partial Checks
The odds of a bank cashing your check depend in part on how and where the damage is. If the tear is confined to one of the corners, you'll have a better shot than if it was shredded into small pieces and taped together. If the amount, account number, signature or the name and address has been ripped, smudged or torn off, the bank may ask you to request a reissued check instead.