Finding the maximum from a two dimensional list

Since you specified "Is there an efficient method?"

GatherBy[Sort@list, First][[All, -1]]

(here list is your list or variable the list is assigned to).

This should outperform the accepted solution by at least an order of magnitude.

Slightly more verbose, but even quicker:

With[{g = GatherBy[list, First]}, 
 Transpose[{g[[All, 1, 1]], Max /@ g[[All, All, 2]]}]]

list = {{1, 2}, {1, 3}, {1, 4}, {2, 2}, {2, 3}, {2, 4}, {2, 5}};
MaximalBy[#, Last] & /@ Split[list, #1[[1]] == #2[[1]] &]

Another one:

data = {{1, 2}, {1, 3}, {1, 4}, {2, 2}, {2, 3}, {2, 4}, {2, 5}};
GroupBy[data, First -> Last, Max]

<|1 -> 4, 2 -> 5|>

If you need the result as a list of pairs:

KeyValueMap[List, %]

{{1, 4}, {2, 5}}

Here's a variation of @sjoerd's answer:

data={{1,2},{1,3},{1,4},{2,2},{2,3},{2,4},{2,5}};

ResourceFunction["GroupByList"][data[[All,2]],data[[All,1]],Max]

<|1 -> 4, 2 -> 5|>

For large datasets, this will be much faster than GroupBy:

data = RandomInteger[10000, {10^6, 2}];
r1=ResourceFunction["GroupByList"][data[[All,2]], data[[All,1]], Max];//AbsoluteTiming
r2=GroupBy[data, First->Last, Max];//AbsoluteTiming

r1===r2

{0.10384, Null}

{0.514609, Null}

True

This approach is also slightly faster than @ciao's.

Similar in spirit to Sjoerd Smit's answer: the wonderful thing about Mathematica's associations is that they automatically override earlier values in favor of later ones, which is exactly what you want! To demonstrate, let me shuffle your input:

data = {{1, 2}, {1, 4}, {2, 2}, {2, 4}, {2, 5}, {1, 3}, {2, 3}}

Converting into an association only keeps the last occurrence of each first coordinate:

AssociationThread @@ Transpose@data

returns <|1 -> 3, 2 -> 3|>. So we just need to sort first (fortunately Sort automatically checks second coordinates after it checks first coordinates):

AssociationThread @@ Transpose@Sort@data

returns <|1 -> 4, 2 -> 5|>. And if you want the answer as ordered pairs again, there's a function that does that:

KeyValueMap[List, AssociationThread @@ Transpose@Sort@data]

returns {{1, 4}, {2, 5}}.