A Virtual Private Network is a network that is constructed to allow connecting to a private network from a machine not physically present on the network. This allows connecting to say, a private company network, over the internet instead of being physically connected to their network. A connection is formed over an encrypted tunnel to the private network, often emulating creating a separate network interface on the users computer to allow the connection to work as if it were physically connected.
A common use case for a VPN is using a VPN provider to encrypt all network traffic through their connection, to protect online privacy and identity. During the connection process, a VPN sets up an encrypted tunnel for communication to take place through - meaning that all data both to and from your connection is encrypted, before it is sent and after it is received. This means that your data is protected from eavesdropping not just from say, people on the network you're connecting through (such as a public WiFi network), but even from your internet service provider inspecting what packets they are transferring for you.
When it comes to online privacy, a VPN provider shifts where you must place your trust - instead of trusting anyone else on your network (such as WiFi eavesdroppers or router-level inspection), as well as your internet service provider and any intermediaries between them and the destination of your connection (a connection being anything from a web page request to a streaming video connection), you instead must only trust your VPN provider. For this reason, many VPN providers have policies such as not keeping connection/data logs, and try to be transparent about how their network operates.