If they’ve been arrested for a state crime, you’ll eventually be able to find them on the Harris County Jail search page here.
It takes many hours for the booking process to complete and for an arrestee to be findable on the jail’s website, so don’t worry if you don’t see any information right away.
You can also call the jail directly at 713-755-5000 (be prepared to be on hold for a while) or visit the Joint Processing Center in person at 701 N. San Jacinto St., Houston, TX 77002.