Depending on the financial institution, the requirements vary some. But, yes you often can open Savings, Checking, Brokerage, etc.. accounts online. The company will often have a process online for creating a new account. But, they also often require customers to personally come into a local Branch and sign a signature card and verify paperwork and collect Identificatio and Contact info from the customer.
However once you have one new account they may let you create/open new accounts without further viists to the branch. Also, they will sometimes let the new customer complete the application online, print out the completed applicaton and then fax it to them signed or drop it off at the branch.
You'll need to call the specific financial institution in question and ask them what they require.