Although there are nuances, at base braising is merely cooking low and slow in a lot of liquid. Very similar to stewing, in fact.
Have you ever done a pot roast? Essentially, that's braising.
As a rule, you first brown the protein. Remove it and brown any veggies you're using (usually, but not always, aromatics like carrots, onions, celery, and garlic). Return the meat to the pot, on top of the vegetables. Then add liquid, coming 2/3 to 3/4 the height of the meat. Cover tightly and cook over low heat until done---which, depending on the protein, can be several hours.
Many people braise in an oven on very low heat. Obviously, that won't work for you.
I'm a little concerned about your comments than most of your cookbook recipes involve an oven. Given the diversity of cooking techniques that only use a cooktop burner, I' surprised to hear that. Perhaps you've not reading them carefully enough? Or you need to find a cookbook that deals more with techniques than with recipes?
Among the many techniques you can try which do not require an oven or other equipment you don't have:
Moist heat: Braising, stewing, poaching, steaming, simmering, boiling.
Frying: Sauteeing, pan frying, deep frying, stir frying
And, of course, combinations of them.
With just the one burner, you'll want to explore the world of one-pot cooking. But you're not confined to it. Just plan out the stages so you never need more than the one burner at a time. And keep in mind that the microwave, because of it's insulation, can serve as a make-shift warming oven.
For instance, say you want to make gnocci in a brown-butter/sage sauce. First cook the gnocci (boiling). Drain them, transfer to a bowl, and put the bowl in the microwave. Then make the sauce, transfer the gnocci (they should still be on the warm side) to the skillet, and gently reheat them. Same thing would apply to anything that calls for a pan sauce.
Give some thought to how you can maximize the space and equipment you have. As Butzy points out, starches tend to stay warm a long time when properly insulated. But you can use non-cook ingredients to advantage as well. F'rinstance, instead of rice, which requires a burner, consider cous cous (which doesn't) instead.