Here is a picture taken from the Dragon Lover's Guide to Pern. As you can see, there are two large straps that surround the dragon's neck, between two ridges. These would be padded heavily with suede or fleece. There are multiple handles up each, to aid in mounting as well as giving a rider multiple places to secure pouches and things.
picture taken from The Dragonlover's Guide to Pern by Jody Lynn Nye
The rider then wear's an almost belt like apparatus that is hooked in turn to the dragon's collars. As your dragon grows, you would add a third large strap so that passengers could strap in as well.

Ok, there's the quick run down of /what/. Now how? :)

A rider isn't expected to be an expert leather worker, but they are expected to be comfortable with the upkeep and simple creation of straps. After all, would /you/ trust your life to someone else? Exactly. Riders want to know the straps are made correctly! Besides it's much easier to fit a dragon when you can talk to it.

So during your Weyrlinghood, you would be given brief lessons on leather working, covering such things as; choosing leather, stitching leather, applying buckles, lining leather. A lot of your spare time would now be used working leather with oil to get the leather supple, you'd knead it, rub it, some have sworn chewing it works best! Whatever it takes! Have fun with the RP! After all you don't want stiff straps to bind or chafe your lifemate.

Now you'd take lengths of knotted rope to measure your lifemate. Climbing here and there, dangling appropriately while he/she tells you where it'd feel best and if it's too loose, too tight etc These measurements would be transferred to your leather diligently with a stylus and for your first few tries, they would be checked by the WLM Staff.

Sewing would be next. Piecing the straps together, as well as edging them to prevent wear. This would be done with an awl and leather cords. Double stitching is the norm. Remember to oil and continue to work the leather, keeping it smooth and supple. After the piecing is completed, lining would be next. Make sure you confer with your lifemate to prevent problems! You'd hate to complete a set of straps that are now too small for him!

Now repeat! Yep, repeat! A second pair of straps is standard. After all, are you not going to fly because you have to repair your single pair of straps! Also you would constantly work on your straps, checking them for wear and tear, worn spots, dry spots *those could crack in between* and all around fit.

Straps are replaced completely at least once a year throughout the dragon's life span. So trust me, you'll get to be old hand at this!