There can only be one registry for each top level domain. If there is more than one index, confusion would result (as has happened to a limited extent with the .biz top level domain).
Registries are run in many different ways. Some are government departments (e.g. the registry for the Vatican www.nic.va). Some are co-operatives of internet service providers (such as DENIC www.nic.de) or not-for profit companies (such as Nominet UK www.nic.uk). Others are commercial organisations, such as the US regstry (www.nic.us). For certain repressive countries, control over the registry and ISPs can effectively control entirely what access their citizens have to the Internet.
Registries can only operate if the top level domain they run has been delegated to them by IANA.
A registry has two main tasks: 1) giving out domain names under their top level domain to those who ask for them; and 2) making the database of domain name registrations available to the world at large.
Registries operate all sorts of systems in order to hand out names. Generally registries operate a 'first-come-first-served' system of allocation. Some registries sell the names directly and others rely on ISPs or registrars to sell them. All registries will have rules about which domain names can be registered. Some of these rules are technical, and so universal, but many are cultural, or depend on the nature of the registry. For example, registries differ hugely in their attitude to obscene or libellous domain names.
The level of charges depends on the nature of the Registry - commercial registries naturally tend to charge what the market will bear, whereas non-commercial registries tend to charge less.
Registries may also impose a system of second level domains on users. The argument for such domains is that it allows more space and certainty in the system. Thus e.g. governemental organisations cannot be impersonated and individuals can have a domain name separate to that given to companies. The argument against is that it leads to less memorable names and fragments the system.
The contrasting approach can be seen in three of Europe's biggest registries. For example, DENIC, the registry for Germany (.de) does not impose second level domains. AFNIC, the registry for France (.fr) has some second level domains, but not all registrants have to use them, and Nominet UK, the registry for the United Kingdom (.uk) requires all names to have a second level domain.
Registries make the index available to the world via Whois systems and via their name servers, for the direction of internet traffic. Such systems have to be fully redundant because loss of name servers can affect all internet traffic sent to that domain.