How are vaccines related to memory?

The memory you are referring to is the "memory" of the immunesystem. Commonly vaccines are injection of antigenes that belong to various dangerous viral/bacterial and/or parasitic infections.

When the antigenes are injected into the body it causes an immune response, various cells related to the immune system (lymphocytes) will attack the antigenes, and antibodies holding that specific antigene will be formed.

This memory can last more or less a lifetime, and when the organism that owns that antigene the immunesystem will recognize it and the organism will be helpless against the immuneresponse and antibodies that are in store for it.

It should also be noted that many parasites have the ability to "coat" themselves, and thus they do not have any specific antigenes, resulting in dangerous parasitic infections that the immunesystem can not fight. (Like for example malaria.)