If you’re interested in a more Hindu-inspired approach, kundalini, energy practices, or the combination of the techniques you might run into in a yoga class with meditation and a broader philosophical system, aypsite.org seems to be a good resource, and /r/kundalini is working on building up a library of experience.
If you feel like your chosen technique hasn’t been covered here and you know a good place to start, please add it here! Message the mods for write access to the FAQ and add it yourself.
Why does meditation have a religious aspect?
The quick answer:
Mindfulness-based stress reduction and mindfulness-based cognitive therapy are probably the driest, most scientific approaches. They were both developed recently with specific goals to be as secular as possible, useful in clinical environments and amenable to scientific study.
The slightly longer answer:
Meditation only made its first big inroads in the West around the 1960s. In India, Burma, Tibet and surrounding areas, the same kinds of meditation that are coming available to us now have been practiced for thousands of years. The culture (and religion!) that’s had time to develop around these practices gives a rich context for describing what you’re doing, where you expect to go with it and what’s happening along the way. For example, sometimes a Pali or Sanskrit word for some meditative phenomenon has a direct translation into English, but using the untranslated word can skip some of the normal English connotations that distract from the core of the experience you’re describing. So that’s one reason you’ll see words like “vipassana” used instead of “insight”, “samatha” instead of “concentration” and so on.
A deeper problem for many people who are just starting to explore this is that lots of explanations and discussions come with more explicit religious baggage. If the teacher you’re trying to listen to is explicitly Buddhist or Hindu, and you’re not, it can be a huge turnoff when they take a detour into talking about the cycle of death and rebirth, karma from past lives and that kind of thing. Just know that the techniques they’re talking about almost always have to do with the present, not with some unknowable metaphysical existence, so you can follow their directions even if you don’t agree with them about the source of our salvation or what happens after death or whatever. They may put things in terms of energy flow or something that you find similarly bogus, but you’re free to suspend disbelief for a few minutes, try their technique and go about your day treating what they said as nothing more than a rule of thumb to help you use your mind in a particular way. This doesn’t have to come at the expense of healthy skepticism.