The first difference between the slice serve and the flat serve is the location of the toss. The mechanics of the toss are the same: hold the tennis ball in your fingers, etc. If you are a little hazy on those details I suggest you review the fundamentals section of the serve. What you’re trying to do is put the toss a little bit further to the side. Not much, just a little bit. That will allow you to swing across the ball later in the serve. This means that for right handers the toss needs to be a little bit further to the right and for left handers the toss needs to be a little bit further to the left.
In the video at 45 seconds we split screen my flat serve and my slice serve. When we play these two clips, you can see that I place the toss a little bit further to my left on the slice serve so that, when we freeze the two clips at contact, I make contact with the tennis ball further to my left than I do on my flat serve. The distance into the tennis court that I’ve tossed the ball is pretty much the same because you do go after your slice serve – you try and hit it hard (typically).