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Actually, I think you should relax and take your advisor's advice. Collaboration is a good thing, and it is a two-way street. You give a bit and you get a bit. I assume you got an acknowledgement in the paper for your help. I don't think it would be appropriate if you weren't. But authorship is a different thing.

You contributed ideas. Research seminars are often organized to give ideas to researchers but the members don't become co-authors in the normal case.

Congratulate your colleague and, as your advisor suggests, spend your effort on your own work, not raising an objection to someone else's.

But, it is good that you contributed ideas. Do that a lot and you will have a lot of people willing and happy to work with you. Occasionally you may need that help.

Pay it forward.

I am planing in taking it to the head of department.

Let me assure you that this is a bad idea. 99 times out of 100 the department head will not intervene in these matters. Moreover even if he does (again, super unlikely), and you get your way about this paper, I assure you that this will forever mar your relationship with your advisor. I would be extremely upset if one of my students went over my head like this.

How about you have a discussion with your advisor about how you feel? What constitutes author worthy contribution highly varies between research groups, so I would not be so quick to decide that your contribution suffices (nor am I in a better position than your advisor to make this call). Maybe you can be more involved in follow ups? In relating this work to your own? As Buffy mentions, you’re not just letting it go because it’s the way it is, it’s also because being adversarial will have far reaching repercussions beyond this one paper!

In my experience, a collaborative approach pays dividends in the long run: be the person people want to talk research with!

When the PhD student started to write a paper about it, I asked him to be on the paper, since I felt I contributed directly to his research. But he said I just gave him suggestions and he could add me in the acknowledgments, not as co-author. And he said I should ask my supervisor if I deserve to be on the paper.

As a general rule, contributing ideas, providing feedback, inspiring others does not qualify a person for authorship. During such discussions, we think and state our own views and ideas without paying much heed to the legwork involved for implementing the same.

Making those ideas come to life are an entirely different thing.

One week later, they submitted the paper to a major conference without my name on it. I felt really insulting that they used my time and my skills with no benefit for me.

You have gained plenty from these discussions. You got to parade your knowledge and mind before others. You draw inspiration from them and later on you may gain a formal collaboration. Also, you gained an acknowledgement. Anything more, would be injustice to the authors involved in the paper.

Science rarely progresses without discussion with people from diverse backgrounds. That is what makes science inter-disciplinary.

In the future I’d suggest being clearer about whether a project is joint work earlier in the process. It doesn’t sound to me from your description that you deserve coauthorship, but I do see why you feel shortchanged. The earlier on that you have this conversation the less chance there is for miscommunication and hurt feelings.

The other answers cover most of the points but I’d just like to add that submission doesn’t necessarily equate with acceptance. They may have rushed this paper slightly to meet a conference deadline and neglected you in the rush to submit. They may be planning a journal submission after the conference and would like to collaborate with you on this. The supervisor/student may have decided to "take a punt" on this conference, knowing that the paper will be reviewed quickly and that reviewers' comments will be useful even if it is a likely rejection.

Bottom line: don’t take it personally. Consider if you want an authorship on a paper that you have had no direct control over.

It's possible that your supervisor and your cuckoo think that you don't deserve co-authorship because they didn't let you edit the paper before submitting it. But the amount of time you spent providing needed advice and guidance in my opinion deserves more than a mere acknowledgement.

That said, raising a stink about this paper isn't going to be helpful to you. You need to finish your own degree, and making an ennemy of your advisor is not the way to get there. In addition, most department chairs are not real managers, and all they'd do with such a request is flee, not try to help you over the opinion of their colleague. So you should probably just let this slide, and maybe try to get an acknowledgement in the final published product.

But you should also pay attention to what this incident says about your advisor. Pursue every possible opportunity to publish with other people! If possible far-away people who don't have any institutional links with him. Be very sure that all help you provide him in your field of expertise will bring you at least co-authorship on whatever the project is, negotiating that up front. Make sure he doesn't just suck you dry for the duration of your studies. Don't outright refuse to cooperate, but be vague and unavailable if necessary.

I have a bi-weekly meeting with my MSc and Ph.D. students that some time is 6 hours long. We discuss ideas and brainstorming solutions; they work on different subjects. When I was doing my Ph.D, I designed an algorithm from A-Z gave it to an MSc student to implement, he published a paper I did not co-author it and did not ask to be a co-author. Why that? because I was a fully-funded Ph.D. student, this is my work I was hired to do that and this was part of my training.

Now, I am a supervisor, and I am on every paper of my MSc and Ph.D. students work. I secure funds for all of them to work on their research.

Are you a fully-funded Ph.D. student, if yes then do not complain you are professional research, not an undergraduate student working on a group assignment if you are not a fully-funded Ph.D. student, then I do not understand why you are doing a Ph.D. Because, if you are that strong and knowledgeable than your supervisor and at the same time not funded, then I strongly suggest that you switch supervisor.

In the future, please tell your supervisor you are busy with your research point and do not go to these meetings.