computer does not have to install SQL Server, unless you want to access the database on computer1 using sql server client like Sql Server Management Studio. In that case you will need to just install the client components on computer2.
You can remote into computer1 from computer2 and do what ever you want to do.
I am not sure what is your application (VB, java etc)accessing the database is written in. The application may need a change on computer2 to access the database on computer1
Definitely you need not have to install SQL Server on computer2