1. register a domain, for example your-domain.com.
2. purchase some web hosting. with hosting, you will get email address using your domain name and also web space to host a website if you need it. emails are accessible worldwide using webmail.
1. and 2. can be of different company or location. you will only use 1. and 2. seperately to get the best price/value. some will do a complete solution (domain and hosting) but i find it is usually more expensive but easier to manage with only one company to deal with.
some will give you 1 free email and limited alias with a domain purchase. so if you only need 1 email, that might be an option. www.1and1.com
it is a bit of a minefield out there for domain and hosting, so make sure you know what you are getting. a .com.sg will need more paperwork to register.