If you cannot even manage to do the half pushups, then what you need to do is the counter pushups, or for the absolute beginners, the wall pushups.
In the counter pushups, you can use the kitchen counter as the name suggests, but certainly you do not use your kitchen space to workout. So, use a chair (make sure its stationary) or a bench or even the bed. Any horizontal surface at the right height would do. You should position yourself on this surface and use that balance to push yourself up and down while balancing on your toes. Like this:
2. Simple Pushup:
And for the ultimate rookie, try doing the wall pushups.
3. Wall Pushup:
To do this, place both your hands on a wall slightly wider than your shoulder width, then kick your legs out and stretch your body till you are on your toes. Push forward towards the wall using the balance of your core and hands, and then push back to the original position to complete one pushup. Repeat this till you are confident enough to move to the counter pushup, then to the half pushups and then to the standard pushups.
Pretty soon, a wall push up for you might start to look something like this:
After a while of doing normal standard pushups, there are an endless number of variations that you can do, like:
4. Ballistic Pushup:
In this pushup, you place yourself either in half pushup or even full pushup position and you lower yourself like the normal pushup but when you come up, you push up with real force and clap your hands and then come back to the ground on both your hands.
5. Diamond Pushup:
Instead of placing your hands at a width wider than your shoulders, here you make a diamond shape with your hands and then do a normal pushup, while balancing yourself on your hands. This will make your body demand more work from your muscles to balance yourself and hence, will give your core a better workout.