What is PBL?
Many methods of solving the 2x2x2 cube finish with the "PBL" step. This stands for "permutate both layers". This basically means that your top and bottom layers are oriented correctly (top and bottom faces are solved), and the pieces just need to be moved around into their correct positions.
There are only 5 unique PBL cases (not including the solved case) if you allow for rotating the cube prior to solving. In practice though, you will want to learn more than 5 cases so that it can be solved without rotating the cube.
But probably more important than executing the cases quickly, is learning to recognize them quickly. At first, PBL recognition can be tricky, but I have some tips for that.
Basic Recognition Method
PBL really just boils down to recognizing 2 unsolved cases on each layer of the cube. These are:
 Adjacent Swap  The layer has 2 pieces sidebyside that are solved, and 2 pieces sidebyside that are swapped.
 Diagonal Swap  The layer has 2 pieces that are swapped diagonally from each other.
So when you combine that across both layers of the cube, you can end up with the following unsolved states:
 One layer solved, adjacent swap on the other
 One layer solved, diagonal swap on the other
 Adjacent swap on one layer, diagonal swap on the other
 Adjacent swap on both layers
 Diagonal swap on both layers
The basic recognition method that most people use is quite simple. First, just see if your top layer is solved, has a diagonal swap, or has an adjacent swap. Then, do the same for the bottom layer.
You can easily tell when a layer is solved, as this should be clearly obvious right away. You can recognize an adjacent swap by looking at all four sides of the layer to see if there is a "bar" of matching colors on any side. Finally, the diagonal swap is when you can't find a bar on any sides.
This method basically involves looking at every side of the cube trying to find the "bar" in each layer if it exists. Somewhat inefficient, but it works when you are first learning!
Algorithms
The left column contains the standard cases, the right column shows the cases from an alternate angle.
Adjacent swap on top: (bar on left)

Adjacent swap on bottom: (bar on left) Same as adjacent swap on top, but cancel the first move with an R2, then add an R2 at the end. 

Diagonal swap on top:

Diagonal swap on bottom: R U R' F2 U' R U R F2 R U' R2 

Adjacent on top, diagonal on bottom: (bar on front) R U' R F2 R' U R' 
Diagonal swap on top, adjacent on bottom: (bar on front) L D' R U2 R' D L' 

Adjacent swap on both layers: (both bars on front) R2 U R2 U2 y' R2 U R2 
Adjacent swap on both layers: (both bars in back) R2 U' R2' U2' y R2 U' R2' 

Diagonal swap on both layers: R2 F2 R2 
2 Sided Recognition
By using some more advanced techniques, you can learn to quickly recognize which case you have by just looking at 2 sides of the cube, eliminating the need to look at all sides.
All you really need to do is look at each of the 2 layers and determine if it has a single color "bar" on any of the four sides. Having a bar on any side of a layer means that there is an adjacent swap on that layer. The only other possibilities are the solved state or a diagonal swap. Solved is obvious just by looking at it, so if you determine that its not solved and doesn't have a bar, then you know that you have a diagonal swap on that layer.
Now, how do you determine if you have a bar on any side? Half of the time, the bar will be on the front two sides, so you can easily see it, but the other half of the time it will be in the back. There is a simple trick to determine when the bar is in the back! Just look at the leftmost visible sticker and the rightmost visible stickerif they are the same color, then there is a bar on one of the two sides that you can't see! So, now you need to determine which of those two sides has the bar. One of the visible sides should have colors that appear on opposite sides of the solved cube, for example red and orange. The bar will be on the side across from those opposite colors. (By the way, if both visible sides of the layer have opposite colors, it's a diagonal swap)
In this example, there is a bar on both the top and bottom layers. For the bottom layer, the red bar is clearly visible on the right side. But for the top layer, the bar is hidden on one of the back sides. We can determine that there is a bar due to the leftmost and rightmost visible stickers on the top layer being the same colorgreen. Then, we look for opposite colors on the visible sides. The right side has blue and green, which are opposite from each other on a solved cube. This means that the bar is located on the opposite side from the blue/green stickers. 
With a bit of practice, you should be able to determine almost instantly whether there is a bar on either layer. Looking for opposite colors to determine where that bar is located is a bit trickier though, but it will also get better with practice.
Trying to switch fully to 2 sided recognition might not be entirely optimal for every person, but having this tool at your disposal can be a big help. For example, 2 sided recognition can help you predict your bottom layer PBL much more quickly during inspection time for an Ortega solve.
2 Sided Recognition Trainer
I have developed a trainer that you can use to practice recognition, here. It simply shows images of PBL cases, and you can try to recognize the case. It doesn't have an answer key or anything, so you need to use the recognition guidelines above to determine the solution for yourself.
Predicting a layer
With some methods of solving the cube, such as the Ortega method, it might be possible to predict the case you will get on the bottom layer, so that you only have to recognize the top layer. With Ortega specifically, if you can plan out where 3 pieces of your first face will go, and you are using OLL algorithms that don't mess up your bottom layer, then you can figure out the bottom layer permutation before you even start solving! If you recognize that you will get a bar on one side, just remember which side it's on, and it's that much less that you have to do during your solve.