What exactly is sparging?

I did my first partial mash about a few weeks ago. I used a midwest supplies kit. The instructions told me to hear up 175 degree sparging water. Now instead of pouring the sparge water in with the actual mash like the directions said I did what death brewer did on his partial mash instructions. I simply took the grain bag out let it drip then placed it in the 170 degree sparge water to rinse the grains and get all of the sugars out of the grains.

Then I did my second batch and I was having some trouble with my boil (which I now fixed). So this time around I did what the directions on the midwest supply kit told me to do. I simply took the 175 degree pot of water and slowly poured it over the grains and let it sit for a few minutes.

Here is my question. What exactly is batch sparging and fly sparging?? Also which one of these situations is the correct way to go?

When I do BIAB, I do it the first way you describe. I've gotten good efficiency, and in my mind at least, it seems to make more sense that way. But in practice the two are probably pretty close.

Sparging is exactly what you just said, using water to rinse the grains and get all the sugars out. I'm not exactly sure what to call DeathBrewer's method, I guess it is a form of batch sparging. Most sparging methods, though, involve draining the water used for mashing, and then pouring the sparge water over the top of the grain, and then draining again. Fly sparging is where you use some type of sprinkler to continuously add water as you drain, and keeping the water level in the tun more or less the same throughout the sparge. Batch sparging is pretty much (gently) dumping all your sparge water into the tun at once and then draining it quickly. With my all-grain batch last weekend I just did the DeathBrewer sparge, so I haven't done a proper batch sparge or a fly sparge. From reading though, it sounds like there are advantages and disadvantages to each method, but if you're just starting out with all-grain batch sparging is probably the way to go since it takes less equipment and requires less attention and effort on the part of the brewer. You'll probably get lower efficiency though.

I can't really speak to BIAB method, but the wiki might answer your questions:

https://www.homebrewtalk.com/wiki/index.php/Sparging

In a traditional mash tun, batch sparging is pouring all your sparge water in at once, stirring, letting sit for about 10-15 minutes, and then draining. Fly sparging is slowly adding your sparge water while slowly draining it out, over the course of about an hour or so. You are trying to maintain about a 1" layer of water over your grains, so you'll slowly rinse them.

For BIAB, your first method isn't really exactly either, the second method is batch sparging. If your efficiency was good with both methods, use whichever one was easier.