The trekking routes up Kilimanjaro require no technical climbing skills, and porters carry gear and supplies on most treks. This makes Kilimanjaro an accessible peak for most reasonably fit people (we met an ecstatic 79 year old man descending from the summit when we were on the mountain). However, all climbers will benefit from some pre-climb fitness training.
You don't need to be an endurance athlete to summit Kilimanjaro, but the better your fitness, the more you will be able to enjoy the experience. Don't leave it til the week before: you should start working up your fitness at least several months in advance.
In the months leading up to your trip, you should plan to do as much walking and hiking as you can. Walk to work, to school, to the grocery store, during your lunch hour... On the weekends you should try to do some longer day hikes (at least 4 hours of walking in a day, preferably more). Wear the hiking boots you're taking to Kili so your feet can get used to them, and you can find out where they cause blisters. If you're not used to walking much, you muscles might be a little sore at first, but better to experience this at home than on the mountain!
If you're fortunate enough to live near mountains, try to log some vertical meters of hiking. You'll be doing a lot of climbing and descending on Kili, and you'll want those muscles to be ready. The climb to the summit might be tough, but descending 13000 feet in a day and a half is really killer on the legs and knees!
Aerobic activities such as running or cycling are good too, but do try to do some longer walks as well. You need to get your body used to extended periods of low to moderate intensity exercise. This type of workout is also good for weight loss -- no point in carrying that spare tire all the way to the summit!
What causes climbers to suffer most on Kilimanjaro is the extreme altitude. Unfortunately there is really no way to train for altitude other than spending time at high elevations. This means it's critical to choose a climbing route that gives you enough time to acclimatize at intermediate altitudes.