Initially, Aeneas is simply a stranger whom Dido welcomes to her city of Carthage, but they end up becoming lovers. Their story ends tragically, however, as Aeneas leaves and Dido commits suicide in her grief.
It is important to note that the gods have pretty much everything to do with this relationship. It is Venus, goddess of love and Aeneas’s mother, who causes Dido to fall in love with Aeneas in the first place, as she wants to ensure that Aeneas is well-treated by Dido. She has reasons to fear the scheming Juno, queen of the gods, going against Aeneas, but then later the two goddesses decide to co-operate in bringing Dido and Aeneas together as lovers. However, the command of Jupiter, king of the gods, overhauls the machinations of these two goddesses as he has a divine, all-important plan for Aeneas to go to Italy, and to establish the future great city of Rome. Therefore he cannot let Aeneas hang around Carthage indefinitely as Dido’s lover, and sends Mercury (messenger of the gods) to remind him of his solemn destiny. So Aeneas abruptly breaks off the relationship and leaves, and Dido is driven to distraction as she’d really believed that they were as good as married.
The tragic love affair between Dido and Aeneas, then, is due entirely to the contrivances of the gods above. However, on the human level, Aeneas does come across as faithless lover, while Dido is elevated to the status of a tragic heroine.
I wanted to add that before Aeneas landed on Carthage, Dido was a strong, independent, and fair ruler. She was diligently building her city, making fair laws, kind of like the efficiency of an ant hill or of a bee hive.
When Aeneas came, Venus realized that she needed to protect her, because he was a foreigner. She sent her son Eros to pose as Aeneas' son Iulus (Ascanius), to make Dido fall in love. Dido became so inflamed by this burning passion for Aeneas that she believed that she had gone against honoring her dead husband. However, Juno wanted to keep Aeneas from ever reaching Italy, so she tried to get Venus to go along with a "marriage" between Aeneas and Dido. Venus realized it was a trick and she played along.
When Aeneas and Dido went hunting in the woods one day, Juno made it storm, so they had to seek cover in a cave. Juno then sent false signs of matrimony (omens of dark things to come), and Aeneas and Dido consummated "their marriage." After that day, Aeneas and Dido were never separated, and Aeneas considered even rebuilding his race at Carthage.
Dido during this time, had forgotten about her image/reputation as a queen, and her city began to fall apart.
Venus approached Jupiter to send Aeneas on his way to fulfill his fate. Jupiter then sent Mercury to tell Aeneas that it was time to go. Dido and Aeneas have a confrontation, and Aeneas leaves for Italy.
Meanwhile, Dido who had lost everything to Aeneas - her name, her reputation, her purity - committed suicide, as she was driven mad by her love and by her guilt.