How to "ignore" an element of Map or MapIndexed

This specific behaviour can be achieved using

If[condition, something, Unevaluated@Sequence[]]& /@ list

The key is Sequence[]. Unevaluated prevents it from disappearing from inside the If.

Alternatively you can use Cases (or many other solutions shown in other answers and comments---some of these solutions may be better suited for the problem but Sequence[] has its place too).

Cases[list, element_ /; condition :> something]

In a similar vein to Szabolcs' first solution, you can use SlotSequence to achieve the same effect.

If[condition, <do something>, ## &[]] & /@ list

You can combine Map (aka /@) and Select:

In[242]:= Sqrt /@ Select[{1, -2, 3, -4, 5}, # > 0 &]
Out[242]= {1, Sqrt[3], Sqrt[5]}