MySQL and MS SQL Server seem to be popular for low cost hosted solutions. If you are hosting the application on your own hardware, the database options increase, but so does the cost. The database itself shouldn't matter outside of setting up your drivers and data source configuration in your application.
As for setting up the schema itself, that is very dependent on your applications and would be pretty difficult to explain in Y!A format. At a minimum, you'll want tables for login credentials. You'll probably want one to maintain the items that are being reviewed and one to contain the actual reviews. If you go with intermediate linking tables, that's an option to consider as well.
"How to build a database (what service, language, etc)?"
- SQL (Structured Query Language)
"Best way to collect the reviews (from a website, someone else's database, etc)?:
- You'll need to write custom collector code to gather from other sites. Unless the site(s) are using a standard format, you'll need a custom collecter per source site.
- From other databases, first the host would need to expose a connection to you. If doing this over the web, I highly doubt they'd give you direct database access so more likely you'll go through some sort of web interface (web service, RESTful service, JSON, etc...).