Most panniers fit most racks BUT beware some dont.
Modern panniers should use a locking, quick-release fitting such a Rixen and Kaul/Ortleib. The hooks are solid moulded plastic and they have an additional part that rotates and loops around the pannier metal rod to secure the pannier. They require a single, round section of metal rod or tube and will not fit racks which have side-by-side sections of welded rod.
Cheaper hooks without the lock may fit twin-section rod but these hooks should be avoided, they can bounce off.

Racks should be bolded to threaded eyelets on the frame at 4 points. If your bike lacks eyelets you can use P-clips or use a rack that cantilevers off the seatpost. Seatpost racks are heavy, weak and can bounce/rotate under heavy loading.

Racks should have 3 legs for strength and stiffness. On some the 3rd leg is bent in a dogleg to prevent panniers hitting the spokes. You need this for cheap panniers but not for good ones. A solid top-plate acts as a kind of fender but if you already have fenders, the open framework top is more versatile. Some panniers have proprietry style of sliding mount for top-bags. I prefer generic style where anything fits. A useful feature is a rear lamp bracket.
Good designs such as Blackburn come in the expensive form and the cheap Chinese copy form. In my experience, both work well. It is the design rather than the craftsmanship that makes a good rack.

Bags should be made of tough, non-flappy material, have a strong, lightweight stiffening in back and base, a heel-cutout profile. Cordura style material will not remain waterproof for years, look for a welded PVC Ortlieb style. Zippers are the first thing to break so many panniers use rolltop or flap-tops which can also be overloaded.