When sentences run consecutively, the defendant serves them back to back (one after the other). When they run concurrently, the defendant serves them at the same time.
Say a jury convicts a defendant of two charges. The judge sentences her to three years in prison for Count 1 and two years in prison for Count 2.
Consecutive sentences. If the judge orders the sentences to run consecutively, the total prison sentence is five years. She'll serve Count 1 for three years and then Count 2 for 2 years.
Concurrent sentences. If the judge were to order that the sentences run concurrently, the defendant's sentence would be three years—she would serve the two-year sentence simultaneously with the three-year sentence.
The amount of time a defendant actually spends behind bars will depend on other factors, such as parole, good time, and supervised release. For more detail on these concepts and the surrounding law, see Concurrent and Consecutive Sentences, and Double Punishment.