Unless you root your device, which can invalidate your warranty with most manufacturers and service providers (at least here in UK - hence why many people don't want to do it), and use a app like ShootMe (which I found to be good and also one of the least intrusive in terms of required permissions), taking screenshots on most Android handsets is a pain.
You can install the Android SDK (and required dependencies) which includes tools you can use to grab screens, but I found it easier for most people to install a tool like MyPhoneExplorer (which is free, includes ADB, and is very easy for non-technical people to install) and the driver for their phone (usually on the handset itself for many devices e.g. HTC phones) instead. The big benefit is not only does it have a built-in and easy way to take screenshots, but also can be used to backup/restore your phone including all contacts/mms/sms/apps/settings/etc, and can be used for multiple handsets (or you can make different profiles for same handset as a crude version control), different phones e.g. Samsung/HTC/Nokia, and different OS e.g. as used on Nokia/SonyEricson/BB/etc, so makes transfer or CRUD of contacts/mms/sms/etc between devices easy and saves on hassles of having different software suites for different phones.
The only solution I have found to take screenshots directly on a mobile without full (permanently) rooting it is to soft root it using the RATC (RageAgainstTheCage) exploit either directly with ADB or far more easily using a tool like UnRevoked. This roots the phone until it is turned off, where upon a restart the phone is once more unrooted.
Lastly, if you just want to take screen shots of your website or web-app, you can do it without any rooting by installing the excellent DolphinHD browser and Screen-Cut plugin. This also has added advantage of being able to take full length shots of your page as one screenshot.
Note: AFAIK this applies to almost all devices I've seen up to Android Froyo (2.2) and most Gingerbread (2.3) and tablets. Some manufacturers may have implemented their own screenshot system.
You can install the Android SDK (and required dependencies) which includes tools you can use to grab screens, but I found it easier for most people to install a tool like MyPhoneExplorer (which is free, includes ADB, and is very easy for non-technical people to install) and the driver for their phone (usually on the handset itself for many devices e.g. HTC phones) instead. The big benefit is not only does it have a built-in and easy way to take screenshots, but also can be used to backup/restore your phone including all contacts/mms/sms/apps/settings/etc, and can be used for multiple handsets (or you can make different profiles for same handset as a crude version control), different phones e.g. Samsung/HTC/Nokia, and different OS e.g. as used on Nokia/SonyEricson/BB/etc, so makes transfer or CRUD of contacts/mms/sms/etc between devices easy and saves on hassles of having different software suites for different phones.
The only solution I have found to take screenshots directly on a mobile without full (permanently) rooting it is to soft root it using the RATC (RageAgainstTheCage) exploit either directly with ADB or far more easily using a tool like UnRevoked. This roots the phone until it is turned off, where upon a restart the phone is once more unrooted.
Lastly, if you just want to take screen shots of your website or web-app, you can do it without any rooting by installing the excellent DolphinHD browser and Screen-Cut plugin. This also has added advantage of being able to take full length shots of your page as one screenshot.
Note: AFAIK this applies to almost all devices I've seen up to Android Froyo (2.2) and most Gingerbread (2.3) and tablets. Some manufacturers may have implemented their own screenshot system.