Witches and wizards employ a number of methods to communicate with each other. Sending letters by owl post is the most common, but other forms of communication include Wizarding Wireless radios and talking through fireplaces.
Trying to contact Harry while he was with the Dursleys during the summer, Ron struggled to use a telephone.
Ghosts can communicate with each other by ghostly letters (CS8).
Fake coins with the Protean Charm cast on them that were used to set up meetings for the DA (OP19). Also, used by Draco Malfoy to communicate with Madam Rosmerta while she was under the Imperius Curse (HBP27).
James Potter and Sirius Black used magical mirrors to communicate during their time at Hogwarts. Sirius bequeathed one mirror to Harry Potter as a way for the two of them to stay in contact, but Harry forgot all about it (OP38, OP24). The mirrors proved their worth years later (DH23).
The original Harry Potter heptalogy was set during the 1990s, so the difference between Muggle and wizarding communication was not quite as dramatic as it is now.