To the question in the title, the answer is yes, but only typically at a later stage of the PhD. The exact timeline varies by field because in some fields students are already publishing early, but a general rule is that conferences are valuable after you have published at least one good publication in the area of the particular conference, and less valuable earlier.
A PhD roughly involves two stages: the first stage is getting matched with an advisor, learning about an area and about one's interests, and progressing along in a project in order to get a good publication. At this stage, attendance at a conference can be fun and interesting, but is typically not that critical. More useful for early PhD students is attendance at summer schools, literature reading groups, and more self-knowledge type of events.
Conferences are not the best for self-knowledge, but much more designed for professional networking and for learning about other cutting edge research in a field where the attendees are already experts. Once you feel you are confident as a researcher in the area and can talk to others in the field without feeling out of place, that is when you get the most value out of conferences. Ask yourself these questions:
If I were watching a talk that was about something I didn't understand, would I be able to ask a meaningful question of the presenter after the talk?
Do I understand the basics of some of the topics that are hot at this conference? Would I be able to discern talks that are related to my interests from those that aren't?
Do I know some of the names of professors working in my area? Would I be able to recognize names and would I feel comfortable introducing myself to those people to chat with them about research?
If the answer to some of these questions is Yes, then attending a conference is good. If the answer to all of them is No, then a conference is likely not the most effective use of your time.
Furthermore, I would like to ask for an opinion of how important participation in conferences can be for a scientific career. Can you do without them or are they indispensable for a PhD-student?
The answer is different for a scientific career and for a PhD student. For the former, conferences are typically very important. They aren't strictly indispensible, but if you don't go to conferences you would have to find some other way of networking effectively and it would be much more challenging. For the latter, PhD students often don't need to attend conferences early on. They are helpful later on, but depending on the level of growth, it could be fine to skip conferences entirely and only attend them later in one's career (e.g. after starting a postdoc).
I would also like to know how to get to participate? Does the chair or my professor organize the participation in conferences? Or should I apply for conferences myself?
You will have to ask your professor about this: you will need funding for travel to attend, so your professor may say that you should only attend once you have a good conference paper to present.
And what is the best way to find conferences that correspond to my field of research?
The best way is to ask your professor and fellow PhD students what are the conferences that people typically attend. Good luck!