Title insurance protects against problems with the legal ownership of your home — a forged deed decades back, an unpaid contractor's lien, a long-lost heir with a claim. Unlike most insurance, it covers the past, not the future: you pay one premium at closing and stay covered for as long as you own the home.
There are two policies. The lender's policy is required and protects only the lender's stake in the property. The owner's policy is optional, protects you, and is usually the one worth insisting on — without it, you could get the keys at closing and still lose the house in court over a defect nobody caught.
A concrete mechanic: before issuing a policy, the title company runs a title search through public records. Most defects get found and fixed then; the insurance covers whatever slipped through. It shows up as a line item in your closing costs, and in many markets you're allowed to shop around for it and save real money.