It's usually rather simplistic to start off by saying "There are two kinds of...", because generally nature, or reality, doesn't come in two well-defined flavors -- like vanilla and chocolate. Instead, there are many flavors, many shades of gray between black and white.
That said, in general I consider there are two views of meditation, dualistic and non-dualistic.
Dualistic approaches typically see us humans as being comprised of an immaterial soul and material body. The goal of meditation is to detach the soul from the body, which enables it to return to god, or merge with god, as in the Hindu conception of atman (soul) and brahman (god).
Non-dualistic approaches typically see us humans as being part and parcel of the natural world, physical beings in a physical world. The goal of meditation is to accept this reality, not being attached to the world being any particular way, but coming to accept things as they are.
After some thirty-five years of embracing dualism, I'm now firmly in the non-dualistic camp. So I've been enjoying a book by Oliver Burkeman, "The Antidote: Happiness for People Who Can't Stand Positive Thinking." Today I finished a chapter about the Buddhist approach to not thinking positively.
Here's some excerpts from the chapter that I liked.
The idea of meditation as a path to calmness is somewhat more realistic, since calmness -- unlike unbroken ecstasy -- can indeed be one of its side effects. Yet all these associations have contributed to a modern image of meditation as a sophisticated form of positive thinking, which is almost the opposite of the truth.
In fact, meditation has little to do with achieving any specific desired state of mind, no matter whether blissful or calm. At Buddhism's core, instead, is an often misunderstood notion that is starkly opposed to most contemporary assumptions about how to be happy, and this places it squarely on the 'negative path' to happiness: non-attachment.
...Rather than merely enjoying pleasurable things during the moments in which they occur, and experiencing the unpleasantness of painful things, we develop the habits of clinging and aversion: we grasp at what we like, trying to hold on to it forever, and push away what we don't like, trying to avoid it at all costs.
Both constitute attachment. Pain is inevitable, from this perspective, but suffering is an optional extra, resulting from our attachments, which represent our attempt to try to deny the unavoidable truth that everything is impermanent.
...Non-attachment need not mean withdrawing from life, or suppressing natural impulses, or engaging in punishing self-denial. It simply means approaching the whole of life -- inner thoughts and emotions, outer events and circumstances -- without clinging or aversion.
To live non-attachedly is to feel impulses, think thoughts, and experience life without becoming hooked by mental narratives about how things 'should' be, or should never be, or should remain forever. The perfectly non-attached Buddhist would be simply, calmly present, and non-judgmentally aware.
Which, let's be frank, isn't going to happen for most of us any time soon.
...When you start meditating it soon becomes apparent that thoughts -- and emotions -- bubble up in much the same uncontrollable, unbidden fashion in which noises reach the ears, smells reach the nose, and so on.
I could no more choose for thoughts not to occur than I could choose not to feel chilly when I was woken by the ringing of the morning bell at five-thirty each day -- or, for that matter, that I could choose not to hear the bell.
...In the analogy most commonly used by contemporary Buddhists, mental activity begins to seem more like weather -- like clouds and sunny spells, rainstorms and blizzards, arising and passing away. The mind, in this analogy, is the sky, and the sky doesn't cling to specific weather conditions, nor try to get rid of the 'bad' ones.
The sky just is.
In this the Buddhists go further than the Stoics, who can sometimes seem rather attached to certain mind-states, especially that of tranquility. The perfect Stoic adapts his or her thinking so as to reman undisturbed by undesirable circumstances; the perfect Buddhist sees thinking itself as just another set of circumstances, to be non-judgmentally observed.