There's 4 types of records that are used commonly but there's a load of other types which you may never run into
An A record is also known as an address record is for IPV4 addresses only and its IPV6 equivalent is an AAAA record. It always points at an IP address.
A Cname, or canonical name record is an alias - it is a record that points at another record.
A mx address is for mailservers and used where the mailserver for a domain has a different ip address than its name.
A txt record is supposed to be for arbitrary text data, but is used for a lot of different things.
Name servers are something that well, tend to work, and unless you had special requirements or choose to run your own, most people don't really notice.
A name server works like a phone directory. When asked for a domain name, it gives the ip address associated with it. Name servers are hierachical - the 'root' name servers know which name servers handle a TLD, which know which servers handle a second level domain and so on. There's two types of name server - authoritative and recursive, but for the purposes of the question, they don't matter.
As for whether to pay for seperate nameserver hosting - your regular host may not have a SLA or have geographically distributed servers. In the case of amazon, you're paying for integration with their services, management tools, and having multiple DNS servers all over the world.