The following situations can occur:
1) The 2 pieces are already in place 2) The top corner is in place but the middle edge is not 3) The middle edge is in place but the top corner is not 4) The top corner and middle edge are on the bottom face
The 4th case can be broken down into 2 distinct cases:
4a) The pieces are next to each other 4b) The pieces are separated.It is possible that a reflection will need to be performed to match one of the diagrams. Reflections will be over the line that cuts the cube diagonally in half between the right and front faces. Therefore, the following substitutions must be made:
R > F' R' > F R2 > F2 F > R' F' > R F2 > R2 L > B' L' > B L2 > B2 B > L' B' > L B2 > L2 D > D' D' > D D2 > D2 U > U' U' > U U2 > U2Note that when both pieces are in place, a reflection or an inversion will both work. Please note that the sequences that start with a twist of the bottom face are enclosed in brackets. This is to signify that the enclosed twist is only when the corner is directly below the place it belongs. If it is not in this position, then it is not necessary to move the corner there and then perform the sequence. Just move the bottom face appropriately so that it will end up in the correct position. In other words, if the corner piece is in the FLD position and the sequence would move the coner from FRD to FLD, it is wasteful to move the piece to FRD and then move it back to FLD.
Before we start, let me emphasize that performing the algorithms with the first layer on the top (cross on the top) is probably the worst option! The images below seem to suggest that you should be performing them this way, but this is not true. When I made this page back in 1997, I thought that showing the cube in this view would give me the best position for the algorithm description and NOTHING else. I perform the F2L algorithms with the white cross in my left palm and heavily use finger shortcuts for the moves. Other cubists proposed to put the cross on the bottom, which is also a good idea. The point is  use whatever seems right for you, but stay away from the cross on the top. This is not a good option.
Pattern  Algorithm(s) 
1) R2 D2 R D R' D R D2 R 

2) R2 D2 F' R2 F D2 R D' R 

3) R' D R' D' B' D B R2 
Pattern  Algorithm(s) 
4) [D] R' D R D2 R' D R 

5a) [D2] R' D B' D' B D' R 5b) R2 D' R2 D' R2 D2 R2 
Pattern  Algorithm(s) 
6) [D] R' D' R D' F D F' 

7) R' D R F D2 F' 
Pattern  Algorithm(s) 
8) [D] F D' F' D' R' D R 
Pattern  Algorithm(s) 
9) F D' F2 R F R' 

10) R' D' R D R' D' R 
Pattern  Algorithm(s)  Pattern  Algorithm(s) 
11) F' R F R' 
14a) F L B D' B' L' F' 14b) R' D R D2 F D F' 

12) [D'] F D2 F' D R' D' R 
15) [D2] R2 D2 R D R' D R2 

13) (D) R' D R D' R' D' R 
16) F D2 F' D' F D F' 
Pattern  Algorithm(s)  Pattern  Algorithm(s) 
17) [D'] F D2 F' D2 F D' F' 
20) [D'] F D' F' D R' D' R 

18) R' D' R 
21) [D2] R' D' R2 F' R' F 

19) [D'] F D F' D2 F D' F' 
22) [D] F D2 F2 R F R' 
1) The corner to be placed is in one of the other 3 top corner positions 2) The edge to be placed is in one of the other 3 middle edge positionsTwist the entire cube so that the piece is in the FR position and perform following 3 twist sequence:
R' D RObviously, reflections can be used so that the cube need not be rotated. Also, be careful NOT to put the matcing piece into the wrong place in the top two rows or this will have to be performed again! For example, you have the middle edge on the bottom layer but the corresponding corner is in one of the other 3 top layer positions. When moving this corner to the bottom layer, don't move the edge from the bottom layer.
To save time, if a piece you need is not currently available, move on to another corner/edge position.