The first that that you need to do is to get into the habit of making the other person feel like he or she is being listened to
. You very well might have the ability to surf the internet or play a game while holding a conversation. The other person might feel that you aren't listening. If it's something important, put down what is in front of you and really listen.
Be sure that while listening to things that you ask questions and repeat it so that the other person knows that you are understanding. Be sure that you don't sound too sarcastic while doing this else it can make the other person feel like you are belittling their efforts.
Make sure that your partner knows that you have the time to listen to him or her. This might not mean that you are doing so right now, but that the door is open for him or her to express the things that need to be said. The idea of knowing one can be listened to can mean alot. This includes keeping multiple options open for the person to get your attention so that the two of you can chat at a deeper level when needed.
From time to time remind your partner about the deeper and more important things that he or she has expressed. This will show that not only you were listening but that you care enough to remember these things longer term. While you can't remember everything do your best to try. The effort will mean quite a bit.