I have the type of personality and skills that would make me suitable for project management, but I only worked independently on many small projects (lots at the same time). I am motivated by thinking of doing this kind of work.

What are people who are assigned to work on projects (at the direction of someone else) called? I have seen job postings that are titled as Project Manager, but they sound more like project worker jobs.

Thanks!

There are a number of accreditation’s you can get, see the Project Management Institute site, or in the UK look up Prince2.
Typically employers look for experience within a particular industry/field though good PM skills are universally transferable.
There are any number of jobs in the Project Management Office (PMO) from very small jobs where the project manger does everything to a PMO with 10’s to 100’s of people managing all the projects within a global corporation.
Roles include administration, change, communications, cost, estimating, planning, QA, reporting, risk/issues.
Like any job the best way to get on is through networking, one avenue may be through popular software look at the Primvera web site