I have been asked that many times over a long long time. It is a tough question because there are so many great digital cameras out there. It also depends on what you can afford. Lengthy discussions can be had by experts as to why one prefers Nikon, the other prefers Canon, and another prefers Olympus, and it goes on. It is always your choice and what you are happy with is the right choice.
Smart phones now can offer excellent cameras. For many that is good enough, it lets them capture moments with a device they already have with them, and are not lugging anything else around. Remember though, picture quality deteriorates rapidly as soon as you start zooming in, try and stay with the widest angle, and move closer if you want to see more. Smartphones offer 'digital' zooms, but that's not the same at all, because these simply crop in on a smaller area of the picture, so you are losing resolution.
The simplest camera is the point and shoot. Cheap, however not much better than a phone! Cheap cameras have cheap lenses, and may give some distortion. These cameras tend to be small, easily fit into a pocket and if all you want is to capture a moment here and there they may be OK.
Long zoom travel cameras. These are a big step up from the point and shoot, have better lens quality and has a much longer zoom range, typically close to 30x. This is also optical zoom not digital zoom. If you can afford it, a long-zoom travel compact is almost certainly a better bet than a cheaper point-and-shoot model. You gain a lot and sacrifice nothing.
Bridge cameras. Size isn't important, but zoom is, then this could be the one for you. The name 'bridge camera' comes from the way these cameras are designed to bridge the gap between a regular compact camera and a DSLR. They have the feel of a DSLR however they don't have the quality because most use the same sized sensors as the smaller compact camera. Some have improved the sensor size, but at a reduction in zoom power. Zoom power can easily be 60x and higher.
High End Compact. These cameras give you higher quality of image, but not the high zoom power.You will get a larger sensor, a better lens, DSLR-style controls and features and sometimes DSLR picture quality. All in a compact size. Many professionals keep one of these as a backup or as a pocket camera just in case! Picture quality is high and you have many options for adjustments.
Digital SLR (DSLR). These are an excellent choice to start your serious photography with. You can learn everything you need to know and they do not cost a fortune. One of the main differences is you can interchange lenses. You get high quality, full manual controls; raw files; APS-C sensor for a big step up in quality. Many of these cameras are very reasonable to purchase.
Mirrorless Cameras. These are just like supersized compact cameras, but with bigger sensors and interchangeable lenses, just like DSLRs. The latest models use new and more sophisticated autofocus systems that put them on a par with DSLRs. They are also more compact and excellent for travelling. You can use the rear screen for composing on a DSLR but it is not convenient. All mirrorless cameras let you compose images on the rear screen with no loss of autofocus performance.
Full Frame Camera.These have sensors twice as large again as APS-C in DSLR's and deliver a further improvement in image quality.These sensors are many times larger than the sensors in the average compact camera and delivers the high quality needed by professional photographers. The full-frame camera's frame is the same size as old 35mm film. The DSLR is smaller.
Expensive, as are the lenses.
So there you are, a very brief synopsis of the main types you would see in most stores.
You need to review what you will be using the camera for, where and your budget. Purchasing used is also an option which can keep the cost down.
Remember one thing. It doesn't matter how much money you spend on a camera, it isn't the camera that composes the image it's you. All the camera does is capture and frame what YOU see,. Learn how to compose the image and once you do, it doesn't matter which camera you get, you will be enjoying awesome images.