What is the most important thing I should know about when hiring a construction company?

get 3 business references from the guy, call them, get copies of the licenses he holds a bond card written bid,,time labor and materials,( provided at cost,no mark up,, time req from start to completion no hourly pay,,,also get 3 proposals,the guy who gets a call and balks about not getting work at the first meeting..let him balk and find someone else

written proposals,references,insurance bond cards

That they are licensed insured and bonded.

Call References that are recent, within 6 months.

Also, references from jobs similar to yours that are from a year or two ago.

Read Reviews.  Ask everyone you know if they can recommend a contractor.

Ask Questions, if communication skills are poor, keep looking for another contractor.

Always have a DETAILED contract with time lines and a payment schedule. If ANY changes are made after construction begins, demand a change order in writting, with signatures from the person who originally drafted the contract, not the workers

NEVER pay in full until you are completely satisfied and the jobsite has been thouroughly cleaned and construction debris hauled away.

Get more than one Quote for the exact same work.

Make sure they construct the kind of stuff you want.  Some construct bridges, some construct tunnels, some construct ships, some construct roads, etc.

. That would be a start.

Ratings from the Better Business Bureau.

if they are going to stay with the job or fit you in between other jobs

First check their expertise, and the work they have done previously. Avoid hiring those if you feel they are not much expert in their field then better to find the one which has much expertise.

Their reputation should be the first thing you check on. You should NOT hire by price alone. Get more than one estimate for the work you want done. Ask questions. If there is ANYthing you don't know, ask. If they give you trouble for asking, cut off the interview immediately and go to the next company. 

A good construction company is glad to answer ALL your questions, has clear and understandable terms, tells you exactly what they will do BEFORE it is done, will give you the most detailed estimate they can (depending on materials and shipping required), and will be happy to give you information about who they've done other jobs for. They show up on time, they complete the work as promised, and they are honest about what they can and cannot do. AND they will be insured or bonded. 

Look at reviews online if there are any--or ask for references. Go see the work they've done or ask for photos. Hire someone who has been in business for awhile. There's a good reason they've been around a while, and it's usually because they do what they say they will. 

if there register with the builder union and state builders register , so if they fail to complete there insurance guarentee there work . nothing worse than a deregistered or not registered builder . ask for customers who recommend you , there reputeable then they will gladly show you ..

I vote the most important thing to know is what you want done.

multiple everything -- bids, references, previous jobs inspected.