1. For watches with spring bars
Nowadays a lot of watch straps are attached to wrist watches with spring bars/spring pins. Spring bars/pins are small metal bars with ends that are bounced.
The first job to do is to take the spring-bar removal tool, and using the forked end, find the end of the spring bar between the strap and the lug (horn) of the watch. You should now be able to push the end of the bar inwards to remove it from the hole in the lug end. You should now be able to pull the strap away form the watch. Repeat this process for both sides of the watch strap.
Next, thread the spring bars through the ends of your new watch strap. Ensuring the watch strap is the correct way around, and that the piece with the buckle is attached to the 12 O'clock' position, slot one end of the spring bar into the small holes on the inside of the lug. Using the forked end of the watch strap, press the other end of the spring bar inwards, before releasing it into the small hole on the other lug. To finish, you should check that the spring bars are properly in the holes and that the strap is securely fastened to the watch so that it won't fall of unexpectedly.
2. For watches with screwed attachments
Some watch straps are attached to watches with screws, such as Panerai. For the most part this is with visible screws on the sides of the watch lugs. For these types of attachments, simple unscrew the screw with a small screwdriver, remove the watch strap, thread the screw through your new strap and secure the screw back in place.