As stated by Sankha, there are only two types of meditation - concentration (samatha) and insight (vipassana). Even these two need not be mutually exclusive.
For concentration meditation, you can have various objects to concentrate on, like the breath (anapanasati), or loving kindness (metta) or unattractiveness (asubha) or any of the ten kasinas. The objective is to cultivate the jhana states.
For insight meditation, breath (anapanasati) is generally used, but the focus here is to observe the nature of reality based on the four foundations of mindfulness. So, even here, it can be on the body, feelings, thoughts or the different mental objects or dhammas, but without an intentional choice of an object of concentration.
Both are not mutually exclusively. For e.g. if you have a camera, you can zoom in and change other settings on your subject of focus, in order to get the best, sharpest and clearest photo of the chosen subject. That's like samatha.
But if you move your camera around to find good subjects to photograph, then that's like vipassana. The four foundations of mindfulness show you what subjects are good to observe, just like in photography you have landscape, animals, portraits etc. Eventually you need to use both techniques to get the best photo of the best subject.
It makes sense to me that learning how to focus your camera first (samatha), is preferred, before learning what kind of subjects to photograph (vipassana). If you try to take photos of your subjects before you learn how to focus your camera, your photos will turn out blurry or too dark or too bright etc (obstruction by the five hindrances). But in the first phase of learning how to focus your camera, you might focus on unimportant subjects like your toes (kasinas).
Another analogy is physics and mathematics. You want to study physics (vipassana) to learn about the nature of reality. You may be very eager to do so. But it would be unpleasant and difficult to learn physics in detail, before mastering math (samatha) first. You need to study math and master it to a certain degree, then you can use it to study physics smoothly and pleasantly.
In the first phase of studying math, you might use it to calculate unimportant things (kasinas) like using algebraic terms like x and y without having any specific meaning, for the sake of performing integration or differentiation on them. When you use it to study physics, then you can differentiate velocity v over time t, to get acceleration a.