Usually, little black insects that locate in kitchens are a type of 'pantry pest' - rice weevils, red flour beetles, grain bores; they are all similar in that they are attracted to foods such as seeds, rice and nuts. You should avoid storing products in paper containers as these worsen the problem - insects can easily penetrate paper bags to get to your food. Be sure to store all of your foods in airtight containers, preferably made of plastic or metal. You should also carefully check products you bring into your kitchen for infestations, avoiding any packages that have been broken.
There is also the possibility that the little black insects are fruit flies. Do they have tiny wings, and seem to be attracted to your sink area in particular? If so, this is probably to case. Fruit flies are (not surprisingly) brought into the household along with fruits such as oranges, apples and strawberries. In the short term, it is advisable to stop buying such fruits until you have dealt with your bug problem. This can easily be done by using appropriate fly killer. Once all the bugs are gone, you can store fruit in your kitchen again - but it may be advisable to buy your fruit elsewhere. Fruit bought from local markets, for instance, can be more prone to infestations of fruit flies, so be careful if this is where you choose to purchase your fruits from.