For companies with less than 50 people, Odoo recommends their Online Edition. This cloud version is hosted on Odoo servers. For this service an extra fee of $25/user/month is added.
For companies with more than 50 people using Odoo, the On-Premise version is recommended. Odoo On-Premise is deployed locally. The Enterprise version of the On-Premise product is more complete than the Community version.