Absolutely. These days, those "dialects" are known as French, Spanish, Portuguese, Italian, Romanian, Catalan, etc.

Of course, we don't refer to those languages as Latin anymore. Latin "proper" can be divided into two major categories: Classical Latin and Vulgar Latin. There are other categories that are used, but they're not particularly relevant to your question, although I might mention them in passing later on.

Latin originated as the language of the Latins, who inhabited Rome and the area around it. The Latin which was spoken until the 1st century BC is often referred to as Old Latin, but strictly speaking it's also a type of Vulgar Latin.

During the late Republic, Classical Latin would emerge as a generally recognized standard for literature. This was the prestige dialect, meaning it was considered "good" or "pure" while vernacular Latin (aka Vulgar Latin) was valorized as base or crude. Generally when people learn Latin in schools today, they learn this language.

Being the prestige dialect, it developed many prescriptive rules governing what was perceived to be "correct" usage. Thus Classical Latin was somewhat frozen in time (there was some change, actually, but generally of a stylistic rather than linguistic bent).

On the other hand, the vernacular speech of the common man continued evolving. This is what is referred to as Vulgar Latin. VL wasn't one singular language across time and space. It had many different dialects across the Empire, some of which would evolve into modern Romance languages. Most VL dialects continued to be mutually comprehensible with Classical Latin and amongst themselves at least until the 3rd century AD. By the 6th century, particularly distant dialects had lost mutual comprehensibility; by the 9th, we begin to see the first writings in recognizably Romance languages: specifically, the Oaths of Strasbourg, which were written in (amongst other languages) Old Gallo-Romance, the predecessor of modern French (and some other regional languages in and around France, like Walloon and Norman).

Language evolution is a gradual process, so the changes I've outlined above didn't happen over night. There is no singular point in time we can definitely say, "Latin ended and Spanish/French/Italian/etc. started," except by choosing a semi arbitrary convention. Thus, (Vulgar) Latin did not strictly die out- it has 800 million native speakers today.

Sources:

From Latin to Romance in Sound Charts by Peter Boyd-Bowman (1980)

Vulgar Latin by József Herman