Depending on how deep or where it's located, they sometimes use glue instead of stitches. 

And depending on what did the cut, you may need a tetanus shot.  They'll probably send you home with an antibiotic and a mild pain killer. 

You wait, you wait---then a nurse or intern will glance at it, ask you how you did it...for animal involvement--they ask about rabies...for metal or ground--when was your last tetanus shot.

They'll clean out any crap in it.  Disinfect it.   

If you do stitches, they'll do a local to numb it, then sew each layer of skin.   You'll feel the tug while they do it.

They'll give you two prescriptions (antibotics/painkiller).

Tell you to keep it clean and dry.

Either the stitches will dissolve or they'll tell  you to get your own doctor to take them out.

All of this takes two hours to be seen, actual treatment is done in less than 8 minutes.

Easier to go to urgent care.

I think Tirzah covered it well, except they will ask for your insurance information!!!

Also there was no   'There there there!'   that helps immensely.

(As is my luck I would often get the newbie doctor who couldn't get the needle in the vein.)
If any one wants know the full SP on what happens when you cut off two fingers with a table saw then I'm your very chap.

Oh and the deep cut they would use either a dissolving suture or permanent one, depending on the estimated wound repair time.
The dissolving suture would be used for sure on the inside skin layer  They used a layer of Bactigras on my wounds, this is a bacterial barrier thin mesh; it's impregnated with a mild anti-biotic; basically nothing will grow on it, so it protects the wound and stops it growing onto the dressing.

Now I'm not sure about what happens in America, but here in new Zealand, a doctor's effort in suturing is proportional to whether the wound will leave a scar, and where the would is. So it is very likely they would only use one layer of sutures on a mans hand. The real find micro sutures are for special cases, it takes a lot more time and equipment.  When they sewed my fingers back on it took them sixteen hours...

Cheers me dears

D A Reynolds

Depends where it's located, they will cover it where they can.