run quickly over a shorter distance or do a slow jog for a longer amount of time? Or both?!

Two factors here.
1) How many caloreis you are going to burn. 2) What mix of fat/free glucose you are going to burn.
Your body burns the same calories covering a set distance no matter the speed. The difference comes after you are done running. The faster you run the longer your metabolism stays elevated.
The other side of this is that the less calories you are burning the higher percentage of that burn is fat. You burn the highest percentage of fat when you are sleeping.
So, first make sure you are lifting as well as running. The muscles you build are what continue to burn calories while you are resting.
I would recommend that you do run walk combinations for weight loss. Start out walking 2 or 3 minutes. Run for 1 minute at a pace that allows you to carry on a conversation, then walk for 30 seconds. After you can keep this up for 30 minutes, up the run to 1.5 minutes, then 2, then 5minutes with a 1 minute walk, the add a minute of running until you are running for 10 minutes, then start lowering the 1 min down by 15 second increments.
You also really need to eat healthier. Balanced meals. Broccoli, Spinach and oatmeal are magical at helping to loss weight.

Basically:
The weight loss is = to the distance you travel, whether you run fast, slowly or walk.
Though you feel very tired after a quick distance run, you in fact burn more energy throughout a longer distance jog.

If you are training for weight loss, the best pace to run at is a nice slow jog, at approx 60 - 70% of you max heart rate. The best way to judge this if you don't want to spend money on expensive gadgets is to jog and still be able to chat comfortably to the person with you. That said mixing it up with a good hard fast session will help lead to a healthy, strong heart.

I'm after losing about 7 or 8 pound, so I've upped my distance and slowed down to 750 mile pace. I know it works because I've done it before. I've had a couple of months off running getting back is why I'm losing the weight. Best thing to do is DON'T WORRY ENJOY

Another thing to think about is that you don't usually start to really burn fat until you've been doing cardio for 20min. Glycogen stores last for 20 min, so the 1st 20 min of cardio is usually not burning any fat.

longer distance for a longer amount of time

I assume your just starting the whole running thing
the best advice to new runners would be:
a) For first week/two run for 30 seconds with a 90s walking time
b) Second/Third run for 45 seconds 60s walking
c) Seriously buy Good running/jogging shoes, you'll get so many pains in your feet if you dont
Each week there after increase the amount of running and decrease the amount of walking........(But don't feel preassurised to jump a level, go at your own pace)......Over time you will greatly improve your stamina

the effort does not really matter. It is the miles. If you run very fast you finish faster so the benefit is the same over the same distance if you were to run slower and take longer. It is about 100 calories per mile. If all you are interested in is weight loss then the benefits are the same.

I would say do both and for the shorter runs try doing intervals. You can try different intensity levels where you increase the intensity each minute from a 5 to a 9 on a 10 point scale. Then go back to a 5 and repeat. Or you can go hard for 1 min or 30 sec and jog for 30 sec.
If you are trying to lose weight, you are going to find more success if you also lift weights 2 - 3 times per week. The more muscle you build, the more calories you are going at rest.
Running is important for losing weight, but I would say that weight lifting is even more important. Combine the two with sensible nutrition and you will have success.

Fartlek - Swedish term for running / walking / jogging. Set distances or time then run flat out, then walk, then jog then run. It's good for stamina and weight loss