Is psychology a science? I've read some books from behavioral economy to emotional intelligence, and it kind of makes sense, but when it comes to psychology it feels to me a bit like astrology, where there are some things that can work, but mostly because of auto-suggestion, or any other kind of suggestions, and things like the placebo (or even the nocebo) effect. So, I wonder if psychology is a science and why so many people rely on it.
"Science" refers to a methodology for obtaining knowledge, and often to the knowledge itself as well.  Science is often confused with another term "technology", that refers to the application of such knowledge for practical uses.  Some people might incorrectly refer to "computers" and "cars" as examples of "science", when in fact they are examples of "technology".

Psychology is similarly divided.  The science part is called "research psychology", or just "psychology".  The technology part is called "applied psychology", that includes psychotherapy, and I think that's what this question is really about.

Clinical psychology is not technically a science, but an applied science, or in other words, a technology.  If you are wondering how science-based different schools of psychotherapy are, then check out this related question: Which schools of psychotherapy are most credible to a hard scientist?.  If you've seen a psychoanalyst recently for example, and you think it's kinda hoaky, well then, you're probably right - psychoanalysis is not generally considered to be science-based, and is comparable to astrology (pseudoscience) in that sense.

On the other hand, other schools of psychotherapy have a more scientific basis: Cognitive-behavioural therapy has been shown to be efficacious (above placebo) for a wide variety of common disorders, and behavioural therapy bases all its techniques directly on scientific findings.