Date today = new Date();
Date tomorrow = new Date(today.getTime() + (1000 * 60 * 60 * 24));

Date has a constructor using the milliseconds since the UNIX-epoch. the getTime()-method gives you that value. So adding the milliseconds for a day, does the trick. If you want to do such manipulations regularly I recommend to define constants for the values.

Important hint: That is not correct in all cases. Read the WARNING comment, below.

UPDATE (May 2021): This is a really outdated answer for old, old Java. For Java 8 and above, see https://stackoverflow.com/a/20906602/314283

Java does appear to be well behind the eight-ball compared to C#. This utility method shows the way to do in Java SE 6 using the Calendar.add method (presumably the only easy way).

public class DateUtil
{
    public static Date addDays(Date date, int days)
    {
        Calendar cal = Calendar.getInstance();
        cal.setTime(date);
        cal.add(Calendar.DATE, days); //minus number would decrement the days
        return cal.getTime();
    }
}

To add one day, per the question asked, call it as follows:

String sourceDate = "2012-02-29";
SimpleDateFormat format = new SimpleDateFormat("yyyy-MM-dd");
Date myDate = format.parse(sourceDate);
myDate = DateUtil.addDays(myDate, 1);

If you do not like the math in the solution from Tony Ennis

Date someDate = new Date(); // Or whatever
Date dayAfter = new Date(someDate.getTime() + TimeUnit.DAYS.toMillis( 1 ));

But more or less since finding this Q/A, I have been using JodaTime, instead, and have recently switched to the new DateTime in Java 8 (which inspired by but not copied from Joda - thanks @BasilBourqueless for pointing this out).

Java 8

In Java 8, almost all time-based classes have a .plusDays() method making this task trivial:

LocalDateTime.now()  .plusDays(1);
LocalDate.now()      .plusDays(1);
ZonedDateTime.now()  .plusDays(1);
Duration.ofDays(1)   .plusDays(1);
Period.ofYears(1)    .plusDays(1);
OffsetTime.now()     .plus(1, ChronoUnit.DAYS);
OffsetDateTime.now() .plus(1, ChronoUnit.DAYS);
Instant.now()        .plus(1, ChronoUnit.DAYS);

Java 8 also added classes and methods to interoperate between the (now) legacy Date and Calendar etc. and the new DateTime classes, which are most certainly the better choice for all new development.