To philander is to flirt or casually date a large number of people. If you philander, you don't have one steady, serious relationship, but instead dabble in many casual ones.
It's most common to say that a man philanders when he is excessively flirtatious or cheats on his partner with many other people. The word has both a sense of sneakiness and lightheartedness, and it comes from the now-obsolete noun philander, "husband" or "lover." The name Philander, in fact, was popular in 18th century literature for the character of a lover, from the Greek word philandros, "with love for the people."