Both web servers and local servers are just computers with software installed on them.
Not much more.
That said a web server is a computer that is connected to the internet and has some kind of a web server software installed on it.
The most common one is an HTTP server software that can serve web pages.
For example, Apache, Nginx are both HTTP servers that can serve both static and dynamic web pages to browser across the world.
Another web server can be FTP, IRC, NTP, SMTP/IMAP/POP3 (mail servers) are all web servers that just have different software installed on them and so they serve other purposes.
A local server is again a computer that serves a client within the local network or LAN.
That means that in most cases it will not be connected to the internet or if it does it will be protected with a password so not everybody can access its services.
It can act as a file server or LDAP server that are roles of a typical local server but it can also be a local web server that holds web-based application only for the local organization.
For example, a company will have a local web server with Salesforce installed on it to serve it’s CRM needs.
To make a long story short both servers are just computers connected to a network. Local servers are connected to the LAN and Web Servers are connected to the WAN. Other than that it really depends on the software you install on them and the use you want to make of them.
If you need more clarification, leave a comment and I’ll try to help.