At its core, dbt (data build tool) is a "modern" data modeling framework using (Jinja) templated SQL. It comes with a CLI tool that allows to easily materialize your models as tables or views in your data warehouse.
The company behind it: dbt Labs (formerly known as Fishtown Analytics) also produces a cloud-managed service revolving around the core dbt. They call it, quite simply, "dbt Cloud". Now, we won't get into that part here, just a thing to note: they've been central to the current "modern data stack" movement, for example coining new job titles like "analytics engineer" and, doing so, they've coalesced 😉 quite a following.
What I mean (and think most people mean) by "modern data stack" is any service (open-sourced or not) that is native to cloud data warehouses (things like Snowflake, BigQuery and co...) and specifically targeting data teams.
What dbt allows is quite useful for data teams. Simply put, we're now easily able to adopt best practices software engineers take for granted (things like code versioning and reviewing, sandbox environments, testing, documentation, etc.) throughout our data warehouses and data modeling processes.