What is a forwarding zone in DNS?

I saw many tutorials and articles talking about what a forwarding zone is. They all say something like, "Use the ForwardingZone option to define an IP address or hostname in which you want mail delivered."
However, the information is unclear. The official explanation of a forward zone is here: "Forward zone - This can be used forwarding one or more mail messages to another address with an SMTP server located within the forward zone." That's all it says. It does not mention what an IP address or hostname means, and the term forwarding also makes me confused. For example, I am hosting a website under a domain name www.com and set up a email to it. Is that "mailing"? Or is the IP address just my computer's IP address?
I don't know the official terminology or if there is a 'special' definition but generally forwarding a mail message implies that the message reaches the mail server for sending but there is no explicit knowledge or confirmation that it was sent and received at the final destination. With forwarding zones in DNS this is the same but they additionally include all steps needed to reach that final destination so the actual destination is explicitly defined. An example would be the following MX record: MX 10 aspmx. This is exactly what the tutorial you linked uses, and it does not use any specific terminology, simply stating that you should define an IP or hostname within the forward zone.

What is an example of a forward lookup zone?

The simplest example I could come up with is the example shown in the DNS tutorial: IN A 1.

1 IN MX 2.2 IN TXT text. IN TXT other-text. The has an A record and MX record for that domain. It also has one TXT record for each "thing" that can have a record of itself (eg, MX records, IP addresses, etc.). The forward lookup zones in these examples are a simple example of a forward lookup zone.

However, there are many other types of forward lookup zones, eg, forward lookup zones that are built out of other forward lookup zones. In the example you give the forward lookup zones are: A forward lookup zone for a subdomain (which is what the .name TLD does) is called an rDNS forward lookup zone.