I googled it, but I can't really understand it.

Wikipedia says: In music, a counter-melody (often countermelody) is a sequence of notes, perceived as a melody, written to be played simultaneously with a more prominent lead melody: a secondary melody played in counterpoint with the primary melody.

It's hard to know what it exactly is. I need help! Could somebody give me examples and explain it?

https://www.youtube.com/watch?v=HeFBrO-fy6o
here's a good example, at 0:23 notice how there's the vocal melody and the piano one mixed together, but one doesn't overshadow the other. Counter melodies are a really nice my tip is to not overdo it cus it can sound messy (^.^)

Here is another (non-EDM) example that I really like, because it really sets the mood for the chorus, at about 0:40: https://www.youtube.com/watch?v=HWFtUbYiCMw. The piano and horn play an independent melody that is slightly more complex than the vocal melody, yet it has a supporting role. Notice also how the melodies join to the same rhythm at points, and then act more independently at others.

EDIT:grammar

One way to write melodies that work together this way is to keep each voice playing different lines of the same chords.  The most basic example would be to have two melodies, one playing the 3rd and the other playing the 5th of the chord.  Now of course, those would just be the exact same melody, but if you start swapping notes between the two or choosing out of chord notes to pass between, then you'll have two melodies that generally work together.  Basically you're taking chords and various inversions of chords, and ripping up the 'chord voices' into melodies.  I can try to elaborate if that's not clear.