Also, please do not listen to how people talk about cover letters on this site. Cover letters are important, and in many industries you can be damn sure they'll be read. And no, they're not 'fanfic about working in your office', or just re-stating the resume contents in a fancier guise. If you treat them like that, then of course they're not doing much for you. That makes you sound like a syncophantic windbag, and it's not doing anything else to help.
I was trained in the dark arts of the cover letter by my mother, who is a hiring manager. And her training has paid off: I've been consistently told my cover letters are excellent. Let me pass on the wisdom.
The job of a cover letter is to give a prospective employer strictly more information than is on the resume. This is where you take the resume contents and put them into context. You look at the job requirements (you are looking at the job requirements before applying, right?) and look at your resume and ask the question (in your head) "why would a person think this candidate would be good for this job?" and your cover letter is the answer to that question. If you don't answer that question, the hiring manager is going to have to do it themselves, and you have way more information to work with than they do.
This is especially important if you're moving from one industry to another, because then the relevant connections aren't obvious. It's also very important if you're overqualified, because then you can use the letter to say "I know it looks like this job would be too entry-level for me, but I'm interested to pursue it for reasons x, y, and z". You can also use this space to explain away something that might otherwise be disqualifying, like job-hopping or a long break in employment. The tone to strike here is always matter-of-fact and constrained to topic - don't go into detail about health stuff or personal tragedy, just tell them what they need to hear: that this period of your resume was affected by [bad thing] but that it's now in the past and you're looking forward to the next chapter. Remember, one of the things they're trying to learn is whether you'd be content in this role, or whether you'd be miserable and leave after 2 months, which would force them to do all the work of hiring again. You're trying to establish that you will fit right in and not cause problems. Knowing what drew you to this job in particular (and don't just say 'money', that's true for everyone) helps with that.
And the other mistake? Everyone writes cover letters that are too long. One page is fine. Two is the absolute maximum, and that's if you have a lot of specific things to say. It's a waste of your time and theirs to pad it out with fluff. Of my two most recent successful cover letters (i.e. jobs for which I was interviewed) one was 390 words and the other was 550. That's the length of this post. How long did it take me to write this post? About 15 minutes to write, 5 to proof and edit. That's how long you should be spending on a cover letter, maximum.
At age 33 I was a mid-career professional in a profession I hated (grant management), and I needed to get the hell out of there. I decided to look for receptionist positions because I understand and basically enjoy computery office work and it seemed like a good way to take a bit of a breather while I figured out what to do with my life. But I knew that if I just sent in resumes, people would take one look at my resume and trash it, because why would a grant manager be applying for an entry level receptionist position, she must be desperate or bad at her job or apt to leave in two months when she finds the job she REALLY wants.
So I wrote my cover letters anticipating those questions, describing my desire to move out of grant management while also describing the transferable skills from grant management that would be useful as a receptionist, and at the end, my reasons for wanting to work at whatever specific org I was applying to (I admire your mission, I have been impacted positively by similar organizations in the past, whatever it is).
I got the first job I applied for, at a public space management nonprofit, and have now been promoted twice in five years and I love my job. And I got it because I wrote a good cover letter to go along with my disqualifying-in-a-vacuum resume.
