What is a sandbox org? What are the different types of sandboxes in Salesforce?

Developer Sandbox
Developer sandboxes are special configuration sandboxes intended for coding and testing by a single developer. Multiple users can log into a single Developer sandbox, but their primary purpose is to provide an environment in which changes under active development can be isolated until they’re ready to be shared. Just like Developer Pro sandboxes, Developer sandboxes copy all application and configuration information to the sandbox. Developer sandboxes are limited to 200 MB of test or sample data, which is enough for many development and testing tasks. You can refresh a Developer sandbox once per day.
REFRESH LIMIT    :-  Daily
DATA LIMIT          :-  200MB

Developer Pro Sandbox
The main difference between this and Developer is the amount of data that can be stored.  It also grabs some product data from production.  If those two things are important, use this one.  Otherwise, it's interchangeble with Developer
Developer Pro sandboxes copy all of your production organization's reports, dashboards, price books, products, apps, and customizations under Setup, but exclude all of your organization's standard and custom object records, documents, and attachments. Creating a Developer Pro sandbox can decrease the time it takes to create or refresh a sandbox from several hours to just a few minutes, but it can only include up to 1 GB of data. You can refresh a Developer Pro sandbox once per day
REFRESH LIMIT    :-  Daily
DATA LIMIT          :-  1GB

Partial Copy
Partial Data sandboxes include all of your organization’s metadata and add a selected amount of your production organization's data that you define using a sandbox template. A Partial Data sandbox is a Developer sandbox plus the data you define in a sandbox template. It includes the reports, dashboards, price books, products, apps, and customizations under Setup (including all of your metadata). Additionally, as defined by your sandbox template, Partial Data sandboxes can include your organization's standard and custom object records, documents, and attachments up to 5 GB of data and a maximum of 10,000 records per selected object. A Partial Data sandbox is smaller than a Full sandbox and has a shorter refresh interval. You can refresh a Partial Data sandbox every 5 days.
REFRESH LIMIT    :-  5 Days
DATA LIMIT          :-  5GB

Full Sandbox
Full sandboxes copy your entire production organization and all its data, including standard and custom object records, documents, and attachments. You can refresh a Full sandbox every 29 days.
Sandbox templates allow you to pick specific objects and data to copy to your sandbox, so you can control the size and content of each sandbox. Sandbox templates are only available for Partial Data or Full sandboxes.
REFRESH LIMIT    :-  29 Days
DATA LIMIT          :-  Same as Production