Not sure if this is still relevant... but you asked for the How... so I'll give it.
1. You first ask for the portfolio of the artist. See the *range* with which they work to see their capabilities. Vector, raster, traditional, colors, etc. If you're not sure you can always get a friend that knows basics to review their portfolio for you.
The mistake people make often is to fixate on an idea of what they want and then see what *exactly* matches it in the portfolio. What you really want is to see if they have the *capability* to do so.
2. Ask about prices.
This is merely a query phase.
This is overall price and the price for consultation and revisions. If you are demanding to pay 15-20 dollars for the entire art piece, that's usually low balling--so shop around, but look at quality more than price. Often for things like logos where they are your brand, you want to pay that extra money for a good artist and save up rather than go cheap and end up with a less imaginative result.
3. Write up a *detailed* list of things you expect in the final piece. Remember that the artist might have the vision, but if you aren't specific, revisions cost extra.
I'll admit that people being unspecific in what they want in art often burns a lot of time. If you're unspecific, I believe that you're leaving it up to me. If you're not leaving it up to me, but have a ballpark, then figure it out. Indecisiveness costs YOU, the client money. From the artist end it's kind of annoying, but if you're going to pay me for it, go ahead. Flip flop alll you want.
For the parts you don't know or aren't sure of, Write it's up to you.
4. Give that detailed report to the person who is doing the commission and ask if they are capable. Consultation fee kicks in at this point for pros.
5. If you *agree* you can work out a contract together. Generally you are the client, and they are the artist. (Artists, don't put down employee). Find a standard one, put down what the commission was into the contract and work out details of ownership and licensing. Do this BEFORE the picture is made. It usually is 10-30% of the contract up front, with another at the halfway mark, and then they tell you they are finished and hold the art hostage while you pay the final amount. This way neither gets shafted. (Most contract work runs this way). Work out a rough due date taking in account for time for revisions.
6. On Revisions and allowing revisions should be in the original contract. Effective of them creating it and you supplying the idea, they effectively still hold copyright and are licensing you to use it since ideas are not copyrightable. (Your write up is copyrightable, but not the ideas contained in it). If you want to pay for complete rights, that'll jack you up. I heard one pro say he was paid 5,000 dollars for one piece and he said that was cheap. If you make a revision without permission that runs into the derivatives and also copyright violations.
Revisions should be paid for. All revisions. They are putting the time and effort in.
If someone is working a burger stand and the customer forgot to say medium rare instead of well done, the burger employee still gets paid for the person's mistake. If the burger employee makes a mistake on the order and makes it again, they still get paid for fixing the mistake. Art should be the same. (Some people don't get this, so this is more for the general audience)
I'd ask around, look for references, search through concept art and maybe deviant art. The person doesn't specifically have to draw say, horses, they just need basic horse head anatomy, a sense of value (light and shadow) and probably you would want to pay someone to vector the head if it's for a logo. (If you want to use it on a poster or a huge banner in the future, you need it to be able to scale, which means you'd have to pay someone to vector it, which means twice the cost)