A Beginner's Method
to Solve The Cube
Basics
Solving the Cube
 Step 1: Bottom Edges
 Step 2: Bottom Corners
 Step 3: Middle Edges
 Step 4: Top Edges
 Step 5: Solve Edges
 Step 6: Position Corners
 Step 7: Solve Corners
 Done
Extras
Step 5: Solve Last Three Edges
 Solve the last 3 edge cubies. 

Basic Idea
Combine Ymoves with Umoves to solve the remaining three edge cubies.
We choose to solve the greenorange cubie first and see what happens to the remaining two cubies in the top layer.
Details
The tricky bit is that any Ymove will affect three edge cubies simultaneously. There are three cases to consider, and they depend on the number of swaps and the number of flips required to solve the final three edge cubies.
Recall that a single Ymove does two swaps and one flip. Therefore, more than one Ymove may be required to do several flips, and if just one swap is required, we have to do combine the Ymove with additional Umoves.
 Solve the greenorange cubie.
In this example, there is just one Ymove to
do the flip, RUiRiU, and the other two cubies
luckily fall into place.If there are two Ymoves to solve greenorange,
check whether one of them solves it all.If greenorange is in the right position but
flipped, you can solve it with two Ymoves.  Suppose greenorange is solved, but the remaining two cubies are not solved.
There are three cases.
0 swaps and 2 flips.
We need two Ymoves to undo the flips.
Start with Y down the front edge,
finish with Yi sideways to the right.1 swap and 0 flips.
Denoting the sideways Ymove to the left
with X = UFiUiF, we do U X U X U.Doing a Ymove with the swapped cubies
orangeyellow and greenyellow does not help.
However, we can swap blueyellow next to
orangeyellow, and then swap redyellow
next to blueyellow, and the remaining
cubies fall into place.1 swap and 2 flips.
We undo the flips and swap separately as shown
in the previous cases. Order doesn't matter.
Comments
The basic issue solving the last edge cubies is that a Ymove corresponds to two swaps of edge cubies. Several Ymoves will therefore do an even number of swaps. If our scrambled cube happens to require an odd number of edgie swaps, then we can solve most edge cubies with Ymoves, but a single swap remains to be done in the end. However, since a single U amounts to three swaps, which is an odd number, we can find a solution involving an odd number of Umoves.
As it turns out, there are four simple moves based on U and Y (sideways) that can swap the last two cubies. Those are UXUXU, UZUZU, UiXiUiXiUi, UiZiUiZiUi. That is, we don't even have to remember the direction of the Umoves or the sideways Ymoves.
Alternatives
We motivate UXUXU et al. by noting that a Ymove involving the two swapped cubies apparently does not help, so we throw in some Umoves and operate on orangeyellow and greenyellow separately.
Actually, I find an alternative viewpoint based on counting swaps easier to remember and more satisfying as well. Let's take the observation that 1 Ymove does 2 swaps to heart.
If we notice that to solve the final edge cubies requires an odd number of swaps, then we know right away that our 2swap Ymove cannot do it by itself. However, we also know that a single U does 3 swaps. Therefore, faced with a cube that requires an odd number of edgecubie swaps, we can do a single U turn to obtain a cube requiring an even number of swaps.
This fits well to our standard strategy of prepare, Ymove, restore. If we prepare with U and restore with Ui, this does not change the number of swaps. Doing any number of Umoves and restoring with their inverses does not change the number of swaps.
Being aware of the swap count therefore suggests the following alternative strategy for solving the last 5 edge cubies. Before you get started, count the number of swaps required. If the number is even, do nothing. If the number is odd, do a single U turn. Then solve the cube with Uprepare, Ymoves, Urestore, taking care that there always is an even number of U moves, which in fact is our default strategy.
The singleU swapadjustment strategy actually works, since the laws of the cube 'preserve' the swap count. Once the cube is in a state requiring an even number of edgie swaps, we can solve it with an even number of Umoves and some arbitrary number of Ymoves.
Alternatives  Examples
First of all, we can understand how UXUXU works from the point of view of swap counting. In the example above:

1 swap and 0 flips. 
In practice, we may want to do that singleU swapadjustment earlier, say after solving just 1 or 2 of the last 5 edge cubies. The closer we are to solving all 5, the easier it is to count the swaps. Ignoring the swap count may mean that we solve edgies that we have to redo in the end. If we adjust the swap count early, we will never encounter the 1 swap with 0 (or 2) flips configuration.
Here is an example:
 Initially, we count two swaps. Redyellow and orangeyellow are in the solved position (ignoring orientation). The other three could be positioned correctly with two swaps: greenyellow with blueyellow, then blueyellow with greenorange.  

Ignoring swaps.  

Notice that there are 2 swaps initially.  

Adjust swaps after solving one cubie. 
Alternatives  Conclusion
In conclusion, to fully appreciate the Ymove, we want to be aware of its 2swap property. However, if you find this entire swapcount business confusing (or, like me, sometimes make mistakes in counting the swaps), don't worry. Just solve until only one or zero edge swaps remain. If you need one swap, just do a single Uturn and solve again. Elegantly with UXUXU if you remember, or just start over and solve with the standard Ymove strategy.