We use Conga Composer, so I can at least give you our use case.
For example, you have Contracts that get signed between yourself and a client with all the trimmings: customer headers and footers, images of your logo, branding and the wording itself. Now imagine the process of having to constantly edit that document to replace the name of the company signing into that contract, maybe different payment variables, the sort of stuff you'd store in Salesforce.
What Conga Composer does, for us, is using a Contract record and after entering in that information, generate a standard document in a Word or .pdf format that can then be sent off for signing.
You can use it for things like employee particulars (medical information, employee-business contracts, that type off stuff) when they start.
We even use it for checklists between ourselves and contractors (we're a recruitment company), so making sure we've ticked off having received proof of identification, checked we've got their right-to-work-in-[insert your country here] information, that type of stuff.
I guess the point is, rather than having to constantly go to a document and find and replace, you can click a button from a record in Salesforce to generate it for you.
The bonus is that rather than people relying on documents that sit outside of a centralised database (Salesforce) which could go missing, get stored improperly, generally cause headaches if you need it 5 years down the line, it's all there in Salesforce for you, a click away, with all your company branding.
In a nutshell, it's Microsoft Word's Mail Merge in Salesforce.
Incidentally, where I say:
sing a Contract record and after entering in that information, generate a standard document in a Word or .pdf format that can then be sent off for signing.
You might (rightly) say: "Why not use an electronic signature service?" Well as we found out, Docusign, an electronic signature company, work with Conga Composer for their product.