my @matches = grep { $_ =~ /aa/i } @array;

print "Matched:$_\n" for @matches;
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Cheers,
Darren :)

my @matches = grep { $_ =~ /aa/i } @array;

print "Matched:$_\n" for @matches;
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Since the nature of the question itself clearly shows that the OP is a complete newbie, I think it may be worth to add that it's not necessary to pass through an intermediate @matches variable (before he cargo cults such an use), although in other cases it could, and in some others it may be convenient anyway:

print "Matched:$_\n" for grep { $_ =~ /aa/i } @array;
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But wait! Here you have two loops, of which one implicit in grep, whereas one would suffice. Hence a solution like that of fenLisesi may be preferable, but (to the OP!) be warned that it doesn't really matter in terms of performance.

This may be what you're after:

use strict;
use warnings;

my @array = ( "file1:aa" , "file2:bb" , "file3:cc" );

my( @found ) = grep { $array[$_] =~ m/aa/ } 0 .. $#array;
print "Matched in $array[$_]\n" foreach @found;
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Update
I wanted to point out that the difference between my implementation, and McDarren's is that his caches the elements from @array that matched, and mine caches the indices of the elements from @array that matched. It's up to you to decide which is better for you. There are subtle advantages to each.

use strict;
use warnings;

my $MATCH_THIS = q(aa);
my @filenames = qw(file1:aa file2:bb file3:cc file4:AA);

for my $fn (@filenames) {
    print "Matched $fn\n" if $fn =~ /$MATCH_THIS/i;
}
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Matched file1:aa
Matched file4:AA
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Update: Added "the content of" for clarification.
Update: Changed @array to @filenames.