"To search" means to look for something. You usually use it with the preposition "for," as in, "I was looking for my little brother." But you can search a closet (for your favorite sweater), or you can search someone's face (to see if he's telling the truth).
"To research" means to gather information, usually with a specific goal in mind. Scientists research methods of curing cancer. I research books and the Internet for my college classes. "Research" is also a noun. Scientists gather research in order to find a cure for cancer.
Rena's example above where she says "research" means to do it again needs to be clarified. To avoid confusion, if you mean "to search again," you would have to write it as "re-search." So you would write, "After I had looked everywhere else, I re-searched my closet for my missing sweater."