Houston Intercontinental Airport IAH is such a well connected world airport with a wide array of terminals, I figured it had to be connected to the downtown by some means of affordable public transport and luckily I got to try it firsthand just last week.
The Houston area transit system is referred to as Metro. It is Metro bus 102 that services the airport, outlying hotel area and the downtown city transit terminals. The Bus can be taken more or less every half hour departing from Terminal C of IAH and the ride is around 45 minutes. There are many stops along the way, so if you plan on riding all the way to the downtown sit further in the back to be disturbed less. There is some good space for luggage on the bus but larger luggage can be stored underneath. The buses are air conditioned and very comfortable.
The schedule varies in particulars but generally operates from 5am till midnight every day of the week. Bus frequency drops on weekends. My experience of the bus took place on a Saturday. There was construction and traffic at one of the times I chose to ride, so it was close to an hour between buses in addition to the already minimized schedule. To see a full schedule check out the online timetables here for Bush IAH Express.
The cost for a one way ride is $1.25 very inexpensive for getting to the downtown. There are smaller and bigger stops all along the way. If you plan on being in Houston for several days or go there frequently consider getting a Metro Q card which can give you even more affordable fares. Be sure to have exact change however if paying with cash as the driver cannot make change for you and credit card is not an option.
The entire Metro system has just been revamped as of August 2015, there are now more frequent buses and routes may have been altered in some cases, but the IAH Express still looks to be running the same routes and options.