It is also worth noticing that if the pairs consist of approximate numbers rounded values can be used to gather pairs whose first elements are close by some designed value:

In[4]:= realPairs = RandomReal[{1, 10}, {30, 2}]

Out[4]= {{5.21667, 6.47609}, {9.31783, 1.36685}, {1.64424, 
  5.90676}, {1.63877, 4.69312}, {8.20514, 9.13797}, {1.43469, 
  7.53501}, {7.32208, 7.86721}, {9.79302, 3.65095}, {3.37067, 
  1.09812}, {4.44002, 8.87933}, {2.05523, 8.75433}, {1.15523, 
  3.88057}, {1.87054, 2.36079}, {7.08747, 7.347}, {1.53206, 
  2.1205}, {6.06558, 6.74602}, {9.85321, 1.59023}, {8.23373, 
  8.98786}, {2.92257, 5.85836}, {4.70867, 9.13353}, {9.7615, 
  7.22106}, {2.21636, 5.98876}, {9.25288, 9.94328}, {8.56543, 
  6.00081}, {9.32182, 2.52546}, {2.91193, 3.88105}, {2.05716, 
  9.22982}, {2.02069, 6.72177}, {5.95095, 1.592}, {9.22202, 3.78433}}

In[6]:= GatherBy[realPairs, Round[#[[1]], .5] &]

Out[6]= {{{5.21667, 6.47609}}, {{9.31783, 1.36685}, {9.25288, 
   9.94328}, {9.32182, 2.52546}}, {{1.64424, 5.90676}, {1.63877, 
   4.69312}, {1.43469, 7.53501}, {1.53206, 2.1205}}, {{8.20514, 
   9.13797}, {8.23373, 8.98786}}, {{7.32208, 7.86721}}, {{9.79302, 
   3.65095}, {9.85321, 1.59023}, {9.7615, 7.22106}}, {{3.37067, 
   1.09812}}, {{4.44002, 8.87933}, {4.70867, 9.13353}}, {{2.05523, 
   8.75433}, {1.87054, 2.36079}, {2.21636, 5.98876}, {2.05716, 
   9.22982}, {2.02069, 6.72177}}, {{1.15523, 3.88057}}, {{7.08747, 
   7.347}}, {{6.06558, 6.74602}, {5.95095, 1.592}}, {{2.92257, 
   5.85836}, {2.91193, 3.88105}}, {{8.56543, 6.00081}}, {{9.22202, 
   3.78433}}}

You could use GatherBy:

(The pure function just presents the first part of each pair to GatherBy.)

In[2]:= pairs = RandomInteger[{1, 10}, {30, 2}]

Out[2]= {{8, 2}, {9, 6}, {7, 10}, {2, 4}, {9, 9}, {10, 2}, {8, 
  10}, {3, 1}, {10, 2}, {2, 4}, {8, 5}, {1, 1}, {2, 8}, {10, 9}, {10, 
  10}, {5, 7}, {4, 6}, {3, 6}, {1, 6}, {6, 10}, {2, 5}, {3, 4}, {8, 
  10}, {2, 10}, {10, 8}, {4, 8}, {1, 7}, {5, 4}, {6, 4}, {8, 10}}

In[3]:= GatherBy[pairs, #[[1]] &]

Out[3]= {{{8, 2}, {8, 10}, {8, 5}, {8, 10}, {8, 10}}, {{9, 6}, {9, 
   9}}, {{7, 10}}, {{2, 4}, {2, 4}, {2, 8}, {2, 5}, {2, 10}}, {{10, 
   2}, {10, 2}, {10, 9}, {10, 10}, {10, 8}}, {{3, 1}, {3, 6}, {3, 
   4}}, {{1, 1}, {1, 6}, {1, 7}}, {{5, 7}, {5, 4}}, {{4, 6}, {4, 
   8}}, {{6, 10}, {6, 4}}}