Scott has a good list. I'll be a bit more generic: It can be applied to anything and everything.
The typical initial 'package' a designer would put together would be the stationery:
- business card
- accessories (envelopes, invoice, etc.)
And then the digital files in a few formats:
- vector for print work (EPS, AI, or SVG)
- raster for web usage (PSD or PNG typically)
Finally--typically for large clients that have to manage their brand across a wide range of offices and vendors--a Branding Guidelines/Style Guide may be provided as well. Take for example Adobe Corporate Brand Guidelines. This is essentially a manual with instructions for other designers to follow that will be using the logos and brand elements in other work.
Beyond that, the sky is the limit. Because of that, it's a good idea to make sure your visuals can eventually work across different media. It doesn't mean a single logo needs to work in all situations, but ideally you'd develop a set of brand assets that can be interchanged depending on what they'd be applied to.
The various media might be:
- paper (stationery, brochures, faxes, etc.)
- physical products (engraved, screen printed, stamped, etc.)
- signage (cut metal, hand painted, thermo formed plastic, vinyl, etc)
- clothing (embroidered, screen printed, thermal transfer, etc.)
- large formats (billboards, livery, trade show banners etc)
- digital (web, mobile, television, etc)
To accommodate all of these variations you may need to create a few variations of the brand elements. Full color for paper, perhaps. Maybe solid line art version black/white printing, embroidery, etc. Maybe an animated version for online and television.
As for your other question:
Is branding designing and doing printing services or designing alone
'Branding' is a huge term and can mean a lot of different things well beyond just the graphic design part of it.
That said, in general, designer's don't typically physically print the work they design, but they may very well manage the process--meaning the client would pay them directly to then find a printer and the printer would be paid by you the designer. But just as often, the client may manage the printing aspect themselves and just ask you for the files. Both are common.