Voicemail routing can take multiple forms, but they're basically rules you can apply to incoming calls that will automatically route them to voicemail without even causing a ring. For example, if calls come in with Caller ID blocked, those can be routed directly to voicemail. Or if you're simply not into talking to anyone, you can hang out a digital Do Not Disturb sign and route all calls to voicemail, perhaps until you're feeling more social or every day between the hours of 9 PM and 7 AM.
Mobile clients are softphones optimized for a particular mobile OS and for being used in mobile situations. This means they're designed to switch easily between different cell and wireless connections on the fly. This means you can let your employees use whatever the cheapest wireless connection around them happens to be—and often that can be free. They also let your employees use your company's phone system on their own devices.
A VoIP media gateway controller (aka Class 5 Softswitch) works in cooperation with a media gateway (aka IP Business Gateway) and connects the digital media stream, so as to complete the path for voice and data. Gateways include interfaces for connecting to standard PSTN networks. Ethernet interfaces are also included in the modern systems which are specially designed to link calls that are passed via VoIP.
VoIP service simply enables your phone calls to be transmitted and received over your home Internet connection instead of your telephone landline. This is perhaps why some people refer to the service as being an Internet phone service or broadband phone service. This does mean though that you need a high speed Internet service to be able to use VoIP for your phone service.