Dark spots left over after having suffered through acne can be as annoying to deal with as acne itself (if not even more so). These dark spots are areas of hyper pigmentation and are more common in people with darker skin, although you can get these dark spots with pale skin as well. If you have these dark spots, it's best to try and treat them as soon as possible. This goes for any type of acne scar, actually. The longer you wait to treat acne scars, the tougher they are to remove. That said, you can remove even old ones, or at least reduce their appearance to something you're happy with.
The more 'dermatologist approved' method is to purchase some skin lightening cream. What these products do is reduce the amount of melanin that's produced by the skin. Over time this will help to provide the person who's receiving the treatment with a more even skin tone, and lighter complexion. Skin whitening products are fairly effective at treating darks spots on the face, and I would certainly look into them if you are able to afford them. Any beautician or someone else who works closely with this sort of thing should be able to advise you on the best product for your particular skin type. This is more effective then buying something from a supermarket or off the internet.
If you don't want to go that route then you can try a more natural solution. Lemon juice is a very effective bleach for the skin and can help to fade these dark spots. Be careful with lemon juice - sometimes it just is far too strong for some skin types, so do a spot test and go from there. You may want to dilute the juice with some water. You can also use apple cider vinegar to help tone the skin. This can be applied to the whole face, a little like a face wash. You can try both lemons and the vinegar for the best results!
Lightening the skin is perhaps the easiest way to deal with these dark spots on the face. I know they're tough to deal with - but try the suggestions outlined in this article to see if they will make a difference to you. And remember there are plenty more effective acne scar reduction techniques that you can also try. With anything, often you'll have to try several things before you find one that works. Acne scarring - no different!