“Power BI” is an umbrella name for a lot of products, which I explain at Power BI first impressions (also I have a video at Power BI for Office 365 video).  Power BI is an Office 365 marketing offering (see pricing).  Each of the products that comprise Power BI for Office 365 have different requirements:

Power Query: Is an Excel add-in that you download and install.  It requires Excel 2010 SP1 or Excel 2013.  Works for 32-bit or 64-bit, but 64-bit is highly recommended.  On-premise only (does not work with Excel Web App).

Power Pivot: Included with Excel 2013, you just need to enable it.  If you have Excel 2010, you need to download and install it.  Works for 32-bit or 64-bit, but 64-bit is highly recommended.  On-premise only (does not work with Excel Web App, but a Power Pivot Data Model can be read in SharePoint online).

Power View: Included with Excel 2013, you just need to enable it.  Works for 32-bit or 64-bit, but 64-bit is highly recommended.  There is not an add-in version for Excel 2010.  Can also be used through SharePoint: run the SQL Server 2012 install (Business Intelligence edition or Enterprise Edition) and on the Feature Selection page choose “Reporting Services Add-in for SharePoint Products”.  Requires Microsoft SharePoint Server 2010 Enterprise Edition, Reporting Services 2012 in SharePoint integrated mode and Analysis Services.  On-premise (Excel and SharePoint) and cloud version (read-only in SharePoint online or Excel Services).

Power Map: Is an Excel add-in that you download and install.  It requires Excel 2013 – Excel 2010 is not supported.  Works for 32-bit or 64-bit, but 64-bit is highly recommended.  On-premise only (does not work with Excel Web App and can’t upload to SharePoint).

Power BI Sites: If you are accepted to the Power BI for Office 365 preview you will be given a SharePoint Online account.  On this SharePoint Online site you will be able to add a “Power BI sites” application that allows you to share workbooks and use other BI features.  Cloud version only – not available with on-premise SharePoint.

Data Management Gateway: This is a client agent (download) that enables cloud access for on-premises data sources on your server and exposes data from these on-premises data sources as OData feeds for corporate-wide access via Power Query.  Works with Windows 7, Windows 8, Windows Server 2008 R2, and Windows Server 2012.

Power BI for mobile: An app you download.  Works for Windows 8/RT devices.  Cloud version only – not available with on-premise SharePoint (only works with data from SharePoint online).

Power BI Q&A: Included with Power BI SItes.  Cloud version only – not available with on-premise SharePoint.

So because Power BI Sites does not have an on-premise version, for those companies that won’t or can’t use SharePoint Online, they will not be able to access on-premise data sources via the Data Management Gateway, won’t be able to share queries created with Power Query (since the data catalogue is in the cloud), and can’t use Power Q&A or Power BI for mobile, so much of the “power” in “Power BI” is not available to them.

More info:

Understanding PowerPivot and Power View in Microsoft Excel 2013

BI capabilities in Power BI for Office 365, Excel, and SharePoint Online