Regarding SharePoint 2013 the SkyDrive (now OneDrive) term is used around the original MySite and some specific content. Microsoft did a really bad job explaining all the buss around OneDrive (old SkyDrive). If we talk about business we are talking about "OneDrive for Business" (old "SkyDrive Pro"):
Terminology and concepts for OneDrive for Business
OneDrive for Business A personal file storage and synchronization service for business use. Users store, access, and synchronize their files in SharePoint Online or SharePoint Server 2013 with their client workstation.
OneDrive for Business document library A document library on the user's My Site in SharePoint Server 2013. Users access their OneDrive for Business document library by clicking OneDrive in SharePoint Server 2013.
OneDrive A consumer-based file storage service available through your Microsoft account. For more information, see What is a Microsoft account?. OneDrive has no relationship to SharePoint Server 2013. However, if you install Office 2013 and open Windows Explorer, you see a OneDrive folder in the Favorites section. This folder synchronizes with the consumer-based file storage, not your OneDrive for Business storage. Users may download the OneDrive synchronization application to synchronize their OneDrive (consumer-based file storage) with the OneDrive folder in Windows Explorer, without installing Office 2013. For more information, see Download OneDrive synchronization client.
Synchronization (also known as Sync) Process that involves copying, updating, and moving files between a client workstation and the server, or vice versa. In OneDrive for Business, synchronization is performed by the OneDrive for Business Windows Sync client, which is either installed with Office 2013, or is available as a stand-alone version, which may be used with earlier versions of Office (Office 2010, Office 2007, and so on).
Source and more details: http://technet.microsoft.com/en-us/library/dn167720(v=office.15).aspx
To your questions:
- OneDrive is your MySite. The OneDrive feature is only available one your MySite host. Since SP1 for SP2013 you don't need a local MySite anymore you can redirect the MySite to a Office 365 OneDrive for Business. (Announcement: http://blog.onedrive.com/one-place-for-all-your-work-files-introducing-onedrive-for-business/ )
- OneDrive for Business client software comes with Office 2013 or is available for free on the web. There is also a Mac version available. For Windows 8, IOS and Android there are dedicated apps in the corresponding app stores. (Details: http://blog.onedrive.com/one-place-for-all-your-work-files-introducing-onedrive-for-business/ ) You need a SharePoint license to access the SharePoint contents, but there is no dedicated license for the client as this is free.
- During the initial startup will try to autodiscover your MySite host and connect to your OneDrive for Business document library to sync. If this fails you can specify the URL of your MySite host. The user will need the permission (configured in the user profile service ) to leverage the OneDrive feature (and create a personal site on the MySite host).
Listen to Todd Klindts Netcast Number 143. Is trying to make it clear and i think he has done a solid job on it.
Here is a blog post with all the details: