In Azure DevOps or Team City what is a build agent?

I'm fresh in the area of CD/CI and I need to implement it in an old project from my company. From what I have read we have a couple of options like Azure DevOps or Team City, I chose these two options because most of our projects are built in Microsoft technologies.

I have been reading for a while, but I cannot grasp the proper definition of a Build Agent; also, I found this old question, but the answer is unclear:

In Team foundation server what is build agent and controller?

Further, I read different documentation:

• Azure Pipelines agents
• Build Agent

And their definitions are the following ones:

An agent is installable software that runs one job at a time. Microsoft

A TeamCity Build Agent is a piece of software that listens for the commands from the TeamCity server and starts the actual build processes. JetBrains

However, I cannot understand exactly their role or purpose, do they build the Test, UAT and Production pipelines in parallel to see if the compilations were successful? Like here:

Or what do they do? Since the solution has multiple projects inside maybe 8 or 10.

You can take the example that I gave in the comments below:

Let's suppose you create a project in Azure DevOps for your new CRM for a Dentist with a Debug, UAT, and Release environments plus a team of 5 devs. What would represent these agents?

Thanks for any explanation.

It's analogous to a human "agent" who has different skills. Think of Build Agent as computer process that has certain capabilities to perform a build.

Some agents can perform certain jobs (e.g. build Apple specific programs), while other agents are more general purpose. Sometimes a computer can have multiple agents that can work in parallel, other times computers only have a single agent assigned to them.

Edit - Added the following to address additional questions: Agents can be "local" which means they are on the server where the build software (e.g., Azure Pipelines, Bamboo, Team City). They can also be "remote" or on a different computer. A remote agent may be needed to build Apple specific software as this software often needs a Mac to compile.

Extending the human "agent" with different skills analogy, agents can be assigned jobs. So one agent may be assigned building software in your pipeline while another agent is busy handling deployments to different environments. Since each "agent" can only do a single job at a time, more agents can speed up build pipelines by allowing parallel jobs.