Here's a more complete excerpt:
"When bricks were more fragile and less standardized than they are today, builders would alternate “stretchers” (bricks laid lengthwise) with “headers” (bricks with the short side exposed). There were headers every six rows in the building, which Enzweiler took to mean that it was constructed around the beginning of the twentieth century."
I'm not sure why, exactly, alternating between "headers" and "stretchers" would be necessary due to the use of fragile and non-standardized bricks. Were the "headers" used to alleviate pressure in the construction somehow, or to compensate for uneven "stretchers" (i.e., make up extra room in the brick layout due to uneven "stretchers")?
The quote is taken from Jeffrey Toobin's excellent piece on the history of lynching and Bryan Stevenson's work at the Equal Justice Initiative (here's the link). The Susan Enzweiler noted in the quote is a retiree who previously worked in historic preservation; she's helping Stevenson with a project that will memorialize the history of lynching in America.
Say a wall is two bricks thick. Run all the bricks in both walls lengthwise, and you would have two walls next to each other. That's not very strong: the two have to be locked together, so some have to run across, be headers. The pattern of doing that is called the bond. There are various kinds of bonds. English, Flemish...the important thing is that the brick acting as a header, going across the wall, is going to be under tension, and brick ( like concrete) is much better holding up under compression than under tension. Since earlier brick was soft, it was necessary to have lots of headers to take that tension- something like Flemish bond , alternating stretchers and headers in each course, or English bond, with a course of stretchers and a course of headers above. Modern brick is harder, can take more tension, so there can be several layers ( or courses) of stretchers for every one course of headers tying the wall together. And that takes less trouble and time than Flemish or English bond.
Why there would be a problem with "non-standardized" brick is less clear, unless the author meant variation in hardness. As the brick would be fired in a big stack, called a clamp, bricks on the inside would get more heat than others on the outside ,and the less-fired would be softer. But the color of the brick would indicate to the mason how high it had been fired, and he could select accordingly.