Forward lookups: Resolves a name to an IP. example, resolves
Google to an IP address.
Reverse Lookup: Resolves an IP to a name. example, given an IP address you can resolve that back to a name.
Forward lookups happen every time you use your browser for example. You type a URL which your browser takes the hostname portion of that URL and resolves that to an IP address (by way of your OS).
Reverse lookups happen more and more often now-a-days with mail servers where they lookup the reverse of an IP address that is trying to connect to them to determine if they will accept email from them depending on some policy.
A zone is just a "section" of the configuration that you place your forward and reverse tables. It is often organized on a per domain basis for forward lookups and on IP subnet boundries for reverse lookup but that's not a hard and fast rule.