DeleteCases depending on all elements

list = {{10, 5}, {20, 4}, {20, 3}, {10, 6}};

1. You can use the function Internal`ListMin as follows:

- Internal`ListMin[-list]

 {{20, 4}, {10, 6}}

2. You can also use DeleteDuplicates:

DeleteDuplicates[ReverseSort@list, And @@ Thread[GreaterEqual[##]] &]

{{20, 4}, {10, 6}}

3. ... and SequenceReplace:

 SequenceReplace[Sort @ list, 
  {a__} /; (And @@ Thread[LessEqual[a]]) :> Last[{a}]]

{{10, 6}, {20, 4}}

You can sort the list into ascending order using plain old Sort, and then Split it into sublists where the second element is ascending. The Last element of each such sublist will be the one you want to keep:

Last /@ Split[Sort[list], #1[[2]] <= #2[[2]] &]
(* {{10, 6}, {20, 4}} *)

If you need the operation to keep the remaining elements in their original order, you can sort again based on PositionIndex.

SortBy[
 Last /@ Split[Sort[list], #1[[2]] <= #2[[2]] &],
 PositionIndex[list]]
(* {{20, 4}, {10, 6}} *)