Cherry eye is a common problem in young puppies. It is actually enlargement of a lymph node on the back of the third eyelid. Even though it looks scary it poses no immediate threat and most puppies are not bothered by it at all. There are two approaches to dealing with it: medication or surgery. Many vets will begin topical medication therapy first and if the cherry eye does not resolve or if it resolves and reoccurs then consider surgery.
I f you research surgery you will probably find reference to 2 procedures
!. Imbrication or tuck which is not removing the gland but tucking back down on the back of the third eyelid-reocurence is the potential problem with this approach.
2. Removal of the gland which makes the likelihood of reocurence very small but does carry some risk of "dry eye" later in life.
Ask your vet about these procedures and take a look at the video about cherry eye.