Most US undergraduate programs do not require an honor's thesis - it is sufficient to achieve take a specific set of courses over four years, and usually the student must maintain a grade-point average (GPA) that stays above some minimum.
At my undergraduate university, there was an Honor's Program that you could be admitted to that did require an honor's thesis. If you completed your thesis (and maintained an even higher GPA than the standard), the you could graduate "with honors". Only a small fraction of undergraduates were in the honor's program. All graduating students got a Bachelor's degree. Students in the honors program got a Bachelor's degree with honors.
Finally, many undergraduate engineering programs do require a senior project, which is usually a year-long project that is done in a small group with other students and produces some working piece of engineering. This is a required course that must be taken in those programs in order to earn the Bachelor's degree in engineering. Other disciplines may have similar senior projects (a musical recital for music majors, for example).
Edited to add stuff about graduate degrees:
At the Master's degree level, some programs require a Master's thesis, many do not, especially in the sciences. In a non-terminal Master's program, which is part of a doctoral program where you satisfy the requirements for a Master's degree (usually just a set of courses) along the way to earning a PhD, there is almost never a thesis requirement. In many fine-arts programs (music, sculpture, theater, etc.), part of the requirements for a Master's degree is a Master's recital or exhibition.
In a doctoral program, where you are earning a PhD, there are usually two major requirements, once the courses that constitute your non-terminal Masters are completed:
A qualifying exam, where a panel of faculty members ask you a lot of questions to prove that you have broad knowledge of your entire field of study.
A dissertation defense, where you present your doctoral dissertation, which is a comprehensive description of narrowly-focused original research that you have completed under the guidance of a faculty thesis advisor. The dissertation is the written requirement: in the sciences it is often compiled from a series of published journal articles, while in the humanities it is usually book-length and may be published as a book. The defense is where you present it to a panel of faculty members who question you about the work you have done.