Answering this question can be difficult as there are many different meditation techniques which result in varying sensations. Also the same form of meditation can result in a different experience for different people.
Generally though meditation is a deeply relaxing experience. Your sense of self-awareness will become much greater, if you’re performing a meditation focusing on you body for example you will be able to think your way around your body and “warm” the area you are focusing on. When you meditate you are very much “in the moment”. If you’ve ever played sports at a competitive level there may have been times when you felt so focused that you were only aware of the actions your were performing (commonly known as being “in the zone”) – meditation is very much like this only your focus is internal.
The feeling of meditation doesn’t end after your session does. You will find with regular meditation that you feel a lot less anxious and more peaceful than before. Many people also find themselves becoming much more accepting of their own flaws as well as those of people they know.