The copyright law gives music owners the right to perform their work publicly, or to authorize others to perform it publicly. By "publicly," I mean any audience that consists of more than a close group of friends or family. Anything larger than that is considered "public."
So if your video contains copyrighted music, and you put your video up on the internet, or on broadcast TV, or on a cable station, or at a film festival or your company's annual sales meeting, someone needs to license the performing rights. Maybe it's you, maybe not.
Most television broadcasters and cable networks have agreements with licensing organizations like ASCAP, BMI and SESAC that collectively represent hundreds of thousands of composers and publishers. These "Performing Rights Organizations" or "PROs" offer blanket licenses that cover all the music they represent. So if your video is going to be broadcast or cablecast on a station with PRO licenses, you'll be covered by their license.
Sometimes, though, stations or networks will decline to take a license from the PROs. In this case, the stations or networks may ask program producers to obtain the performing rights for all the music in their programs directly from the music owner.
Restaurants, hotels, skating rinks, convention centers and even businesses that play music for customers or even employees also need to make sure they have licenses to perform music.
What about the internet? Technically, performances on the internet are public performances, and need to be licensed. Some of the larger sites have licenses with the PROs, but for smaller sites, you'll want to make sure you have the necessary clearances. Check with your music supplier if you're not sure.
The last thing anyone wants is a phone call about copyright infringement.