Tailcoat is a broad term to describe a type of formal coat that has a longer back section than front section. Essentially the front of a tailcoat ends at or above the waist, and the back splits in to two symmetrical “tails” that hang lower.
Traditionally tailcoats were designed this way to make riding horses more comfortable; though the style became so popular that it stuck around long after riding horses became less mainstream.
Today there are two main styles of tailcoats commonly worn; the more formal "dress coat" (worn to white tie events) , and the morning coat. The dress coat is cut squarely in the front, while the morning coat tapers below the button. These two tailcoats were originally designed to be worn at different times of day, with the morning coat, of course, to be worn in the morning, and the dress coat to be worn in the evening. Today no such distinction exists, and either can be worn at any time of day.
While tailcoats were at one time popular for every day wear, today they are relegated to formal occasions. As Victorian sensibilities started to wane the tailcoat was deemed too cumbersome and not versatile enough for an industrialized society.