I received my first review request while I was a PhD student and the topic was about something similar to the work I did during my master studies. I felt very honoured but also a bit lost during that time. Reviewing manuscripts is nothing you are taught at University (why not???). One of the first questions often ask in this context is: how do journals choose reviewers, do they find you or how does it work? To be regarded as an ‘expert’ in a certain field and become a reviewer, you usually should have some papers published in decent journals, preferably as being a first author. I don’t manage manuscripts, but I guess editors just put some key words of the repective paper in Google Scholar and check who has recently done similar work. If you did and your name pops up, you are certainly interesting for them. Then they might additionally check for your current position, if you are PhD student, Postdoc etc. and for an email adress so that they can reach out to you.
Next they will likely send you an email through the journal’s system including the review request. You usually receive the abstract and name of authors (if review is not double-blind), and then have to decide if you feel capable of providing a review in a certain amount of time. This could be anything between one week and several months depending on the journal. I previously received requests where the manuscript was already attached to the email, and this is a certain sign that the journal is not so decent. I would commend to never accept such requests. If you reject the review, you often have the opportunity to provide a reason. Good reasons to reject are: conflict of interest (you know the authors too well or currently work on a similar topic), you have no time, subject of manuscript is beyond your area of expertise, you are on vacation, sick, or your computer just died. I once heard that for every paper you publish yourself, you should review at least 3 other manuscripts. This gives a good estimate of how many reviews are adequate. I usually like to choose those paper where I know enough about the topic but also get the chance to learn something new (method-wise or similar).
If you accept the offer, you get immediate access to the whole manuscript and can download it. Sometimes you have to register with the journal’s review system, sometimes this is done for you already. It is your responsibility to keep the work confidential. So better don’t forget a hard copy on the office printer or in the bus that everyone can see. Then the review process can start. I usually have a first quick read of the manuscript and highlight some things I don’t understand or have questions about. Then I put the paper aside and come back to it at a later time point doing a more thorough read and writing the report in one go. For the review report, each journal has its own wishes. Most of them ask for a summary (that shows that you actually understood the main message), some specifically ask for the main strengths and weaknesses, quality of the language or novelty of the work. Some provide specific forms, other ask for more ‘freestyle’ reviews, it really depends.
You might ask for the benefits of doing so much unpaid extra work. Well… it’s true there is no monetary benefit. At best, journals provide vouchers or discounts for your own publications. Some journals publish reports thanking their reviewers with your name on it or provide review certificates, which you can print and put on your wall to make your colleagues jealous ; ). You can also sign up for an account with Publons, which allows your reviews to be tracked. After the review is done, you receive a ‘thank you’-email from the journal and can forward it to Publons. Reviewing served me a lot in terms of learning critical reading and thinking, developing ideas, and providing something (my expertise) back to the science community, which is a good thing! Also, after having received some bullshit reviews on my own papers, I get the chance to ‘make it better’ and express the kindness I would expect from others. Therefore I enjoy reviewing papers a lot!