Easy! (though getting the new one in just the right position can take a few tweaks).
There are a few different seatpost clamp designs. Most likely is that there's a curved surface attached to the top of the seatpost and a curved piece that mates to it that sits above your saddle rails. Each of these has serrations in the center, and matching those serrations is how you change the tilt of your saddle. There are slots on the side of each piece that hold your saddle rails. There's a 5 mm Allen bolt that sits between the saddle rails underneath the saddle and goes up through the two parts of the clamp, through a curved spacer/washer, and into a nut. That's what holds everything together and lets you slide the rails forward or backward in the clamp. You may or may not be able to get the old saddle off just by loosening the bolt and rotating the top part of the clamp so it sits between the rails. If you have to take the clamp completely apart to get the old saddle out, then pay attention to how it all went together before you take the bolt all the way out.
There are other possible designs, but that's the most common one. Just look underneath your saddle and take a look how the seatpost clamps onto the rails of the saddle. If it isn't immediately obvious to you how it works, take a picture and post it here.
Put the new saddle on the same way the old one came off. Make sure the clamp is snug but not overtightened. You can do that by feel, unless your new saddle has carbon rails; in that case use a torque wrench.
Eyeballing the position of your old saddle is probably close enough to start with. There are markers on the rails that show the maximum fore and aft positions for the saddle; don't go beyond those (if you have to go beyond those marks to make the bike fit properly, that means you need a different seatpost). If you had a detailed fitting with your old saddle, and/or if you're prone to knee trouble, I can explain how to take a few measurements to get your new saddle as close as possible to the same position as the old one; but that said, every different saddle I've used, even ones that were VERY similar to the ones I took off, has COMPLETELY changed the way I've sat on the bike, so even if you put the new saddle exactly where the old one was, it would likely be just a starting point anyway.
Be aware that you might need to raise or lower your seatpost (using the clamp at the top of your seat tube), since different saddles are different heights. That one also takes a 5 mm Allen wrench, and if your frame and/or seatpost is carbon, use a torque wrench.
Your first few rides with the new saddle, make sure you have your 5 mm hex key accessible so you can change your saddle position if you need to. If you've got carbon bits and need to make a quick adjustment on the road, as long as you properly torqued everything at home, just count the turns of your wrench as you loosen, and make sure you re-tighten everything the same number of turns.
I just took longer to explain it than it will take you to do it.