The domain name is used to establish your identity. It is used in general of « Foundation » to build your reputation.
Choose a meaningful, short and memorable name.
First of all, it takes clear keywords to increase your natural search engine rankings on these.
It should also be taken to choose an extension depending on your strategy and the nature of your business:
- .com if it is commercial or international
- .biz so it is rather in the business
- .fr if it is local
There are a multitude of extensions. You can take that which best fits your business.
About the ‘mobile friendly’, you can either buy the .mobi or simply opt for a prefixed subdomain of the « m » (for example: m.thecowmunitymanager.fr).
Finally, it may be interesting to have multiple domains (.fr, .com) to first improve your visibility and secondly to protect (for example: www.cdiscount.com and www.cdiscount.fr). You will just have to do a 301 redirect.