What is an acquired taste? Some foods are delicious. People love eating them, and the experience has a positive reinforcement. Thus, people will eat the food again.

Some foods are instead "challenging", at least initially. For example, spicy or unfamiliar food. There are some foods which are considered particularly challenging like Durian fruit or Century eggs.

Why don't people simply stop eating this food, and instead acquire a taste for it?
One explanation for the development of 'acquired tastes' is a form of reinforcement learning called flavor-flavor learning. The name sounds a bit odd on its own, but should best be seen in relation to the concept of flavor-nutrient learning.
What flavor-flavor learning is
When a previously unexperienced flavor is encountered with familiar flavors, the eater may learn to associate these tastes. This is called flavor-flavor learning. There is some experimental evidence to suggest that flavor-flavor learning is automatic when eaters are hungry (Brunstrom and Fletcher, 2008). The study is not as well powered as might have been preferred, however, so the chance of false positives should be kept in mind.
How flavor-flavor learning explains taste acquisition
Supposing the Brunstrom and Fletcher result is not a false positive, we then have a plausible idea of how acquired tastes might develop from a flavor-flavor theoretic viewpoint: people are exposed to novel flavors in combination with liked flavors at meal times, either through parents or friends or just eating out. Since we tend to eat when we are hungry, in the long run this induces flavor-flavor learning and hence taste acquisition.
If you want to know more, a forward Scholar search on the Brunstrom and Fletcher paper lists some subsequent studies that deal with acquired tastes in additional detail.
References

Brunstrom, J. M., &amp; Fletcher, H. Z. (2008). Flavour–flavour learning occurs automatically and only in hungry participants. Physiology &amp; behavior, 93(1), 13-19.