If you have not done so already, you can download TouchCopy from here.
TouchCopy will copy voicemail messages from your iPhone if your iPhone supports Visual Voice Mail. These will be voice mails that are provided as part of the Visual Voice Mail data plan from your mobile service provider.
You can determine if your mobile provider includes visual voice mail as part of their service from here: http://support.apple.com/kb/HT1937
TouchCopy will not be able to access voice mails that are stored remotely by your operator, i.e. those that you can hear by dialling your network provider.
Provided your mobile service provider supports visual voice mail, you can then copy those voice mails stored on your iPhone by referring to the TouchCopy user manual.
TouchCopy will save your voicemail messages the format they are stored on your iPhone - as .amr files. You can play back your saved iPhone voicemail messages using the AMR Player software for Windows or QuickTime for Mac.