Rightly so, this question is a regular, mostly due to the fact that DSLR prices have dropped and are within reach of an average person's budget.
I'm going to try and keep this as simple and 'un-technical' as possible.
Here are a few pointers to help you make your decision...
If you are looking for a camera that fits in your pocket then obviously a DSLR is not for you. Besides having a much larger body, be prepared to carry around a couple of lenses and/or accessories.
2. Camera response time
Are you into action or sports photography? If you are irritated by any delay between pressing the button and the photo being taken, then I suggest you seriously consider a DSLR.
3. Night/low light shots
DSLRs have bigger sensors than compact cameras and therefore are more sensitive to light, giving you much better image quality at high ISO levels. Due to the flexibility of manual settings you can also play around with slow shutter speeds to capture the night ambiance and lights without 'spoiling' the image with a flash to obtain more light.
DSLRs open up a world of possibilities with their ability to change lenses. From wide angle lenses to superzooms, to macro lenses to photograph subjects unseen to the naked eye - every lens tells a different story. Add to this a large range of other accessories, such as flashes or different filters, and your DSLR can produce a variety of different images and adapted to a thousand and one situations.
Whilst a point and shoot camera requires no maintenance at all (that is, unless you are careless with it!), a DSLR requires a bit more care. One has to take into consideration that each time you change a lens you run the risk of letting dust into your camera; meaning that your DLSRs image sensor would need to get a professional cleaning every once in a while.
So what should you buy?
Even though I've mentioned a few differences between the two you might still be unsure. Here are a few questions to help you conclude:
Do you want to take more professional looking photos?
Do you want to be able to change the lens attached to your camera?
Are you looking to have more control over the images you are taking?
Do you want to take better photos in low light?
Do you need a camera which responds very quickly?
If you answered 'YES' to all the above questions - then a DSLR is the camera for you.
Buying a DLSR needn't break the bank. Canon's entry level DSLRs start from under €500 including a standard kit lens to get you going.
Having said all this, I own both a point and shoot camera and a DLSR. I like having both as I believe they compliment each other and whilst a DLSR is super flexible and produces amazing top quality photos, my compact comes in handy as it is so convenient and quick to reach for and snap away!
Any questions, please comment below!