You can only upgrade to SharePoint 2013 from SharePoint 2010 so those of you looking to upgrade directly from 2007 will have to take the 2010 route first.
There is no in-place upgrade option for 2013, not that we ever used it, so every upgrade needs to be onto new kit and either done via the DB attach method or by creating a new farm and using third party tools to migrate the content.
- 64 bit Windows Server 2008 R2 SP1 or 64 bit Windows Server 2012.
- 64 bit SQL Server 2008 R2 SP1 or 2012.
- 2012 SP1 is needed if you are planning to use BI
When you move 2010 site collections across to 2013, they remain in 2010 mode and SharePoint 2013 maintains both a 14 hive and a 15 hive to support both 2010 and 2013 mode site collections. You can still view and access 2010 mode site collections but they will only provide the 2010 level of functionality; site collection administrators will get visual warnings on the page when a site is in 2010 mode.
If you have 2007 site collections in your 2010 farm, then you should upgrade them to 2010 mode before moving them to 2013. There is no visual upgrade tool in 2013 like there was in 2010 so to upgrade 2010 site collections to 2013 mode you will have to run a PowerShell command, or by using the Upgrade this Site Collection item in Site Settings (site collection administrators only).
Microsoft recommend that site collection administrators are left to upgrade their own site collections post upgrade, rather than as part of the upgrade. The exceptions of course are those site collections which may need particular attention; such as high volume, highly customised, or critical sites.
Once a site collection has been marked for upgrade to 2013 mode, an item is added to a new upgrade queue which is processed by an Upgrade Site Collection timer job. This timer job runs every minute and can run parallel upgrades; there is a throttle applied to prevent the server being over utilised by this activity.
It is possible to view what a 2010 mode site will look like when it is upgraded to 2013 mode by using the Create an Evaluation Site Collection function. A daily timer job processes this request and copies the 2010 site collection to a new 2013 mode site collection within the same content database, gives it a URL with the same name as the source site collection and appends -eval on the end. By default this is retained for 31 days after which point it will be deleted. The idea is to let a site collection administrator look at the site in 2013 mode and determine what changes will be needed before upgrading the site for real.
There is also a site collection health checker that can be run against both 2010 mode and 2013 mode site collections.