Each state has their own rules, and Moshe's link may provide a comprehensive list of steps (or not). But generally, eviction works like this:
1) The tenant breaches some portion of the lease, most often non-payment of rent.
2) You make demand for the rent. If your lease is written correctly, demand is implied and doesn't have to be made separately.
3) You post any legal notices such as "Pay or Quit" and wait however long your state laws require. Missouri (my state) doesn't require pay or quit. I can file the actual eviction the day after rent is due.
4) The tenant fails to "make good" on the demand for rent during the pay or quit time and then you file for eviction, or unlawful detainer.
5) The court sets a date for a hearing. During this time, usually the tenant can make payment of all funds due and then you would request to dismiss the case or just no one shows up and it is dismissed by failure to appear.
6) You attempt to serve tenant official notice of the hearing, according to your state laws. Some states require personal service (i.e. hand them the paper) by a sheriff, constable, or a private person called a special process server. Often, this step is required if you want a money judgment. If you cannot get service, you may just have to post on the door, which often only leads to you getting possession of the premises. The theory here is that you must INFORM them of what they already know: that rent hasn't been paid.
7) The court date arrives! You go and plead your case and present your evidence and the tenant may or may not show up. Even if the tenant pays before court day, it's probably a good idea to at least show up and withdraw the case so you do not develop a reputation as a sloppy land lord who doesn't care about the court's time.
8) If the tenant DOES show up and wants to argue that they are not delinquent or contests the amount owed, you may end up going to trial. Trial in my state is anywhere between 1-2 months after the initial hearing. I MAY get a judgment for possession before trial happens, but the trial is for how much money is owed. This is not a trial by jury: it's a bench trial where each side swears in, testifies, presents relevant evidence, and the judge makes a decision about two things: possession of premises (if not already awarded) and money judgement owed.
9) Assuming you win at the hearing and/or trial, you get 1 or 2 documents - Doc 1 is a Writ of Possession which you file with the court. Then the Sheriff/Constable will call to arrange a setout date by which time the tenant must be completely out of the property, and if they aren't you can MOVE them and their stuff out and set it on the curb/law. If they resist, the sheriff or constable will try to get them to comply, but if they are belligerent they can be arrested. Doc 2 is the money judgment, which you can try to collect a number of different ways that involve too much detail to go into here.
Good luck! I'd recommend hiring an attorney this first time, even if you don't intend to always do that. Go along and watch, listen, learn how court works. Then maybe try DIY next time!