I teach my friends this:
Step 1: Orient edges of the last layer, done by applying the algorithm R U R' U' R' F R F'. Number of times it needs to be applied can be from 0 to 3. You use it to move from the case with 0 edges oriented to the case with 2 adjacent edges oriented. From there you use it to move to the case with 2 opposite edges oriented by applying it when the 2 edges are in the right and front position. From there you use it to move to the case with 4 edges oriented by applying it when the 2 edges are in the right and left position.
Step 1 more "advanced": Orient edges of last layer by learning the 4 different cases. Case 1: All edges oriented, you're done. Case 2: 2 edges oriented and they are adjacent to each other. Place them in the right and front position and do the algorithm f R U R' U' f' (lowercase f means do a wide F turn, taking the middle layer, the one between the F and B layer with the F layer when you turn). Case 3: 2 edges oriented and they are across from each other. Place them in the right and left position and do the algorithm F R U R' U' F' (basically like case 2 just with a non-wide turn). Case 4: Simply do case 3 and then case 4.
Step 2: Permute corners, done by applying the algorithm R U' L' U R' U' L U2 which switches the position of the back-left corner with the front-left corner. Number of times it needs to be applied can be from 0 to 2. To know where to use it, just turn the top layer until you find 2 corners in their correct place (there will always be either 2 or 4, if you have 4 you don't need to do anything obviously). Once you find them, they can either be across from each other or adjacent. If they are across, just apply the algorithm and there will be an adjacent pair of correct corners. If they are adjacent, put them on your right and apply the algorithm, that should fix all the corners.
Step 3: Orient corners. The algorithms you need are Sune (R U R' U R U2 R') and Anti-Sune (Either do Sune mirrored which is L' U' L U' L' U2 L or do the inverse of Sune which is R U2 R' U' R U' R). Playing around with only these algorithms will eventually orient the corners. You'll get familiar with all the cases with time, the basic idea is to reach a stage where only one corner is oriented and from there all you need is either Sune or Anti-Sune applied in the correct angle. See how Sune and Anti-Sune affect the orientation of the corners to know what they do so you can use them to orient the corners. You can also solve using just Sune or just Anti-Sune but they are not hard to learn and using them both will get you what you want faster.
Step 4: Now that corners are oriented and permuted, and edges are oriented all we have left to solve the last layer is obviously permute the edges. Here you can solve with just 1 algorithm (which needs to be done 0-2 times if you use it well, 0-3 times if you don't). There are 4 different cases of edge permutation (known as the PLL's Ua, Ub, Z and H). The algorithm is R2 U R U R' U' R' U' R' U R'. What it does is cycle the position of the edges, The left one goes to the right, the right one goes to the front and the front one goes to the left. You can play around with it until the cube is solved.
Step 4 more "advanced": You could also use the inverse of the algorithm from before which is R U' R U R U R U' R' U' R2 to cycle the edges in the other direction, which can sometimes help reduce the move count.
The minimal number of algorithms you need to learn for this way to solve the last layer is 1 for each step meaning only 4 algorithms to solve the last layer.
I have no idea what this way is called but this is how I was taught to solve the last layer.