Types of Cameras
When buying a camera (and lens), the main thing you need to look out for is sensor size. Generally the bigger the sensor the more detail can be seen and more background blur can be achieved. The issue is that as sensor size increases, prices also exponentially increase. A full frame sensor, is a 24 x 36 mm sensor and is the same size as a negative strip of film and these sensors are still quite expensive. I would stick to an APS-C sensor (also known as cropped sensor) as they are the most reasonable price to quality ratio for someone starting out.
Full frame 36 x 24mm
APS-C 23.5 x 15.7 mm (may vary between brands)
Micro Four Thirds 17.3 x 12mm
Phone sensor - tiny
Megapixels generally don't matter. In fact, if you squeeze loads of megapixels onto a smaller sensor, the image quality starts to suffer - a larger pixel size can take in more light so there is a cleaner image with better dynamic range (differences between darks and lights). The only time you should pay attention to pixel count is for printing. Generally above 10 megapixels is fine for printing up to A3. I've even done it with 8MP. You only really need 24 MP+ to start printing large format commercial prints such as billboards.
1. Traditional Compact
These are the traditional digital cameras with small sensors (the same size as the phone ones) and have generally been made obsolete by camera phones
2. Bridge Cameras
These are usually cameras that look like dSLRS but have a fixed superzoom lens and a small sensor. Waste of time unless you like spying on very distant people in mediocre quality.
3. Large sensor Compact
So an amazing technological advance is that there are some compact cameras with APS-C sized sensors in. These can be had for cheap and will far outperform any camera phone. These are definitely worth considering if you want a very discreet camera with full control. Generally people that know what they're doing get these as casual shooters. You can even get a full frame compact camera - the Sony RX1R II - but this costs about £2000 and what is the point.
Back in ye olde days, cameras had 2 lenses, one for looking in and one for taking the photo onto the film. These were known as twin lens reflex cameras as seen here.
Then someone had had a stroke of genius decided it would be better to have 1 lens which you looked through and transmitted the light onto the film. Thus the Single Lens Reflex or SLR camera was invented as seen below.
Originally they used film, but when the digital revolution arrived, the film was replaced with electronic sensors and digital single lens reflex cameras or the dSLR was born.
These can have either a crop (APS-C) sensor a full frame senors and were a long time considered the go to choice by professionals. This is changing with mirror less over the last few years. Nevertheless, if you are beginning to learn photography, this is likely the best choice.
These vary enormously in their shape size and make-up. You can get them in all sensor sizes, with a fixed lens (like the large sensor compact mentioned previously), or interchangeable lenses.
Initially the main selling point of them was that they are far more compact as they don't require the mirror mechanism. However, a lot of them that have replaced dSLRs professionally are basically the same as dSLRs expect without a mirror. With the modern ones, the differences become negligible and it comes down to personal preference of what you want to see through the viewfinder - the real world optical view, or a projected digital image. Often the same sensors are in dSLRs and Mirrorless cameras so image quality ends up being the same.