Centering is just simply centering your energy in your body. You can do this several ways, but it's all the same concept. The way I do this is by sitting up straight. I focus on my energy as I breathe. I inhale and send energy to my centre (center is usually from the sacral to the solar plexus. Just experiment to determine exactly where your centre is). Exhale. Inhale again sending your energy to your centre. As you do this consistently your energy forms a ball. You don't have to see your energy doing this movement with visualization, but it helps to. At the very least you need to be able to feel your energy. So, having experience in feeling energy and energy control is a prerequisite. Do this until you feel you are centred efficiently.
Grounding is an extension of centering. The general concept of grounding is being a tree. Your roots go deep into the earth, and excess energy and even negative energy can be grounded out. (If you'd like you can look into grounding concerning circuit boards to get another idea of what grounding is. Just about the same concept). Any ways, after you have centered your energy continue by inhaling. When you inhale you breathe in energy sending it to the energy ball at your centre. Exhale. When you exhale you send energy down your body and out through your root chakra. This energy extends out into the ground. It starts small, but grown thick and long as your continue through this process so that it reaches out like roots of a tree. The roots stabilize you; or helps to. Allows excess energy to be released. A way to push out negative energy. Even a means of pulling in energy through these roots as needed. There are even methods of making a shield that is made by extension of the roots made by grounding.
Both centering and grounding can be done in a few different ways, but they tend to have the same concept (as prior said). You can can do this standing, sitting, or even laying down. But the latter may not be the best way. Just experiment to determine what works best for you.
How to meditate is best determine by you, what your purpose of meditation is, and what works best for you.