BES (Blackberry Enterprise Server) is basically Blackberry's proprietary form of EAS (Exchange ActiveSync) for Microsoft Exchange corporate email servers. It's a conduit that allows mobile phones to get real-time updates of your company Email, Contacts, and Calendar (and Notes on Blackberry devices) from your Exchange email account.
Both BES and EAS allow the Exchange system admin to have some control over the device, such as requiring the user to enter a password to access company email, and the ability to remote-wipe the device if it is lost or stolen.
Only Blackberry phones work with BES. All other devices, can license EAS from Microsoft. EAS is available for iPhone, Android, and many other platforms.
Most wireless providers charge extra for corporate email on Blackberry devices connected via BES (~$15/month added to your data plan). Blackberry subscribers that don't upgrade to BES can still access non-real-time email using BIS (Blackberry Internet Server), but are blocked from corporate/Exchange email access. Non-Blackberry phones such as iPhone and Android, can access corporate Exchange email systems without restriction using EAS on a basic data plan without paying extra.
Depending on the device, you may sacrifice some control and privacy by connecting to Microsoft Exchange. This is by design, as the whole system is designed to ensure accountability for company information. Older phones may not allow you the ability to segregate private from company data, instead mixing all contacts and email into a single data pool which is synced to your company's Exchange server. Most newer devices running iOS3+ and Android will allow you to selectively segregate this data so that you can still keep your private info separate from the company's.
Hope this helps!