Android phones come pre-installed with two email clients "Mail", which you can use for general email accounts, including Exchange servers with Active Sync, and "GMail" which you can use to access your gmail or google app email accounts (gmail for your domain). Apart from these there are some excellent free and paid options which you might want to consider, specifically K9Mail and TouchDown. K9 (a free app) should handle all your advanced email needs, but you might want to consider TouchDown ($20) in case you have problems accessing your corporate email.
"Mail" is an email client that is built-in into the Android OS. It can connect to POP, IMAP and Exchange servers that support ActiveSync. It won't connect to Exchange servers that allow OWA (Outlook Web Access) access only. This is a basic email client that could satisfy the basic emailing needs, but if you think something is missing from it (e.g. signatures, etc), then you should try K9 (see below), which is an open source project based on this stock application.
GMail is the built-in email client that can be used to access your Gmail or Google mail for your domain. It flawlessly supports multiple accounts and can push new emails to your mobile. This is a great option if you primarily use gmail for your email needs.
K9 is an opensource email client focused on making it easy to chew through large volumes of email. This email client is based on the built-in (stock) email client ("Mail"), but with countless improvements. It provides excellent access to IMAP and Exchange (OWA) accounts, including support for multiple inboxes so for example it can notify you of new messages in multiple IMAP folders. Apart from the multi-folder sync, K9 supports email signatures, bcc-to-self, configurable sender address, return address configuration, keyboard shortcuts, message flagging, IMAP deletes, saving attachments, configurable notifications and more.
The K9 client is free and can be installed from the Android Market.
TochDown from Nitrodesk lets you access your corporate email, contacts, calendar and tasks right on your Android device. It supports Exchange, Zimbra, Kerio, Novell GroupWise and other servers used in corporate environments. If you can't access your corporate email account with other clients listed on this page, try TouchDown, it will probably work flawlessly. This is specially the case with corporate Exchange servers that only support OWA access without ActiveSync.
The application costs $20, but it's well worth the price. You can try it out free for 30 days to ensure that you can connect to your account. Nitrodesk allows you to buy the licence directly from them in case your SIM card is from one of the many countries that are not allowed to buy apps on the Android Market.
ProfiMail is a powerful email client that supports POP, IMAP and SMTP. This is a port of the app that was a great success on the Symbian platform (S60 3rd edition), so it might be a good option for users moving from Symbian to Android that used it on Symbian. The list of features is extensive and includes support for multiple accounts, push email (using IMAP IDLE), signatures, HTML email, support for various character encodings (Western, Cyrillic, Central European, etc) and excellent support for attachments. Price: $27.99, can be bought directly from the vendor.