Le Cinque Terre, the five lands, are a group of five picturesque villages along the coast surrounded by vineyards, olive groves, and woods. The villages can be reached on the train that runs between La Spezia and Genoa or by ferry from La Spezia, Portovenere, Levanto (the next village up the coast toward Genoa where there's also a train station), or other Italian Riviera villages.
There are popular hiking trails between the villages as well as in the scenic hills above them. You'll have to pay for entrance to the trails. Europe for Visitors has a good Cinque Terre hiking guide and map with pictures so you'll know what to expect when you go.
The Cinque Terre region is very popular with Americans and is very crowded in summer. Spending the night in one of the villages is a good way to experience the charm without the huge crowds since they aren't packed with hotels. My friends recommend staying in Manarola at A Casa Rooms and Apartments. Vernazza would be another good choice.