The concept of a morning suit goes back almost two hundred years. Although its a British originated idea, its still a good idea to know about it.
The original idea came about toward the end of the 1700s in England. It was actually developed as a more athletic wear and was very popular with riders. The long coat was eventually trimmed to become the tailcoat pattern that we see today, which is the dead giveaway that a guy is in morning dress. Its still very popular at horse races in the UK, as well as daytime weddings and in some other formal events.
In addition to the coat, many men also will wear a tophat with this, especially if its a more formal occassion. You'll also see ascots and a turndown or wing collar shirt. Grey is a very popular color although you'll also see black in a more formal occassion. Most people will say that a morning suit is typically all gray while black with a tophat is the more formal morning dress. Its a little bit of splitting hairs I think.
Recently it came up in the news because the of the royal wedding between Prince William and Kate Middleton. Because the wedding was taking place during the day, it was assumed that morning dress would be the protocol. Most wedding-going men did oblige, although it was rumored that the prime minister himself would wear a business suit, although he did not. Needless to say, the British are a little more fussy about things like this.
It was also popular in the 1930s to the 1950s through the movies of Fred Astaire. It gave him a dashing look especially while dancing and soon was copied by many men in daily life.