You can run MobileFirst Server either on-prem, meaning you install it locally in production-ready server that you maintain yourself,
Or you can use IBM Bluemix. In Bluemix you are "given" a space where you can either use the Mobile Foundation service, which takes care of setting up a server for you with MobileFirst Server, or you can go through some steps to create it yourself.
To install on-prem: https://mobilefirstplatform.ibmcloud.com/tutorials/en/foundation/8.0/installation-configuration/
To install on Bluemix: https://mobilefirstplatform.ibmcloud.com/tutorials/en/foundation/8.0/bluemix/
In case of on-prem be sure to first contact IBM to purchase a license so you could gain access to download the required files.
The developer kit is like on-prem, and provides a ready environment for development purposes locally in your computer. It is like the Developer plan when using the Mobile Foundation service plan in Bluemix.
Developer kit: https://mobilefirstplatform.ibmcloud.com/tutorials/en/foundation/8.0/installation-configuration/development/mobilefirst/
Developer plan: https://mobilefirstplatform.ibmcloud.com/tutorials/en/foundation/8.0/bluemix/using-mobile-foundation/
