What is a crosswalk?
It is a portion of the roadway designated for pedestrian crossing.

How are crosswalks marked?
Crosswalks are designated by surface markings, including lines, on the roadway.
Crosswalks are also present at intersections, although they are not always marked.
Who has the right of way at a marked crosswalk - a pedestrian or a vehicle?
Generally, the pedestrian.  The law requires vehicular traffic, on the half of the roadway on which a pedestrian is using a crosswalk, to yield the right of way to the pedestrian. Additionally, vehicular traffic on the opposite side of the roadway, going in the opposite
direction, must yield the right of way to the pedestrian once the pedestrian reaches a point in the crosswalk where he or she is in such proximity to that half of the roadway, as to be in danger.  A pedestrian may not, however, leave a sidewalk or other place of
safety and enter onto a crosswalk and into the path of a vehicle that is so close that it is impossible for the vehicle to stop. 

Essentially, when the pedestrian is in the crosswalk, vehicles must yield the right of way to the pedestrian.  However, a pedestrian may not proceed into a crosswalk, from a safe location, when vehicular traffic is in such proximity that it is impossible for the vehicle to stop.

What about pedestrians that choose to cross roadways at locations other than designated crosswalks?
The pedestrian, then, must yield the right-of-way to vehicular traffic on the roadway.  The Texas Transportation Code (Sec. 552.008), though, requires vehicle operators to exercise due care to avoid hitting pedestrians on the roadway, allows the use of an audible device (horn) to warn pedestrians who are on the roadway, and requires drivers to exercise precaution upon observing a child or "obviously confused or incapacitated person" on a roadway.

Are the laws different for crosswalks at intersections with pedestrian signals?
Yes.  If control signals are present, pedestrians must wait until the signal gives the "walk" signal, before proceeding into the crosswalk.  While in the crosswalk, pursuant to the "walk" signal, the pedestrian has the right-of-way.  A pedestrian may not enter a crosswalk contrary to a "Don’t Walk" or "Wait" signal.  If a pedestrian is in a crosswalk pursuant to a "walk" signal and the signal changes to "Don’t Walk" or "Wait," the pedestrian must proceed to a sidewalk or safety island.