Physicians save lives in the dangerous world of Azeroth by using bandages and antidotes to heal injuries and remove poisons. By turning cloth of varying quality into potent medical supplies, they can quickly get wounded heroes back into a fight, supplementing (but not replacing) magical means of healing.
First Aid is a secondary profession - anyone can train as a physician, no matter how many other professions they've devoted themselves to.
Antidotes and bandages can be created almost anywhere, as long as you have the appropriate ingredients - cloth (for bandages) and venom sacs (for anti-venom).
Make sure that you have learned the plan you'd like to create, and that you have the required cloth and other special materials listed for that pattern. Cloth can be acquired from many enemies (or often from the auction house); venom sacs are found on the corpses of several poisonous creatures, and other materials are available from trade goods vendors.
Once you have the materials you need, you can make bandages and anti-venoms at any time and nearly any location - simply choose 'first aid' from your professions menu, choose the item you'd like to make, and click 'create'.