When you access a website, your web browser is making web requests to a web server to retrieve specific information, like a web page, image etc.
A proxy is a service that sends those requests to other websites on your behalf. This can be used to avoid disclosing the true source of the requests to the destination website (such as your IP address or browser fingerprint) or to avoid disclosing the true destination of your requests from some monitoring or filtering service sitting between the origin and destination (such as a corporate proxy or firewall that tries to block you accessing the destination server). Similar to the reasons people use VPNs, but proxies aren’t as secure.
A residential proxy is a proxy service hosted on a residential internet connection provided by an Internet Service Provider.
A data center proxy is a proxy service hosted in a data center (a big warehouse-type building filled with enterprise computer hardware), which likely does not need to use a commercial ISP (the companies who run data centers are powerful enough to have direct connections with major internet backbones).
Hope that helps!