Drill pipe connections (or rotary shouldered connections to give them their proper name) have the fluid seal is at the shoulder and the thread is to hold the joints together and transmit torque and stress along the drill string. There are a range of API drill string connections, NC50, NC38 and so on (but a lot of people continue to use the old names such as 4-1/2" IF) and a number of proprietry threads such as DSTJ, PAC90 and so on. All you really need to know about drill string connections at the rigsite is what a your connection's make up torque is, and which connections are interchangable. The Baker Tech Facts book will help with the commonly available connections, both API and speciality drill pipe connections from the likes of Prideco. The Engineer will select a particular thread type based on the torsional strength of the connection, but may also consider things like the OD of the connection (will it fit into the hole? Could it be fished?) the ID of the connection (what is the hydraulic performance? Could I run tools through the middle of it, or fish my MWD tools through it?).
Casing and tubing connections come in two main types: API connections (8 round and buttress) which rely on dope to make a seal, and premium connections from a whole host of casing manufacturers that don't and instead have a metal to metal seal, either via a step in the middle of the thread or at the pin. API connections are made up to a marker on the joint, premium connections are made up to a specific torque, and correct make up is checked with a torque- turn graph.
All you really need to know on the rigsite about casing & tubing connections is what the torque rating is (if you think you might have to rotate the casing string) and for premium connections what the required make up torque is and what the make up loss in length is. For Engineering, you need to know the tensile and rotational strength of the connections (it's almost always less than the strength of the pipe) and choose your connection according to the design stresses on the casing and whether you have any corrosive fluids present. The various manfacturers' websites have lots of infomation on their connections and connection selection. Often different brands of casing connections are more common in certain areas: VAM's connections are ubiquitous in the North Sea, whereas Hunting's connections are probably more common in the Gulf of Mexico.
Finally, there are the large diameter threads for top hole casing (36", 20" etc) that are solely designed to be made up quickly (Vetco's RLS4, and so on).