With web hosting, you can create custom email addresses under your domain name e.g. firstname.lastname@example.org, email@example.com. The number of free email addresses, that you can create, is limited. This limit is decided by your provider.
Each email account is assigned a certain space (e.g. default 250 MB). This space is configurable.
There is a limit on the number of incoming and outgoing emails. In case of shared server, the number of outgoing emails is limited to around 60 per hour. If the number exceeds this limit, the account (website) is considered to be spamming and relevant action is taken by the server (e.g. blocking the account).
There exists a limit on the email disk space too. If the space occupied by emails exceeds a pre-defined limit, the provider will ask you to delete useless emails or to opt for a different email server. When such a need arises, you can ask for an email server from the same provider (i.e. your web host provider) or from a different provider if you find its plan more lucrative.