The Voicemail service is a service supported by almost all carriers for recording voice messages when you do not answer an incoming call. Voice messages are not stored on your device but are stored at the carrier, and can be replayed by accessing the service, by holding down the "1" button on the dialing number pad, after opening the Phone app.
The "1" shortcut is a convention that applies generally to all cellphones. Behind the scenes, using it causes your device to dial the special Voicemail number of your carrier. This needs to have been programmed into your phone.
Allows choosing the voicemail service you wish to use. In many cases you won't have a choice, and this option won't be used.
The Voicemail number is the number your phone uses when dialing the traditional Voicemail service of your carrier. In many cases your phone will be supplied pre-programmed with the correct number or will pick it up from your carrier, and you won't have to alter this.
This setting might be located under a Setup menu for your chosen Voicemail service.
Allows choosing a sound that will play when you receive new Voicemail. This can be disabled by choosing the "Default notification sound" and setting that sound to "Silent" in your Sound settings, or by reducing the volume of notifications to zero, if your device has a separate volume control for notifications.
Enables or disables a vibration alert when you receive a new Voicemail.
For carriers that support this feature, enabing this allows you to use Visual Voicemail, a way of seeing and playing Voicemail messages using menus on your screen instead of a traditional touch-tone phone interface.
Your carrier must not only support this feature, but do so for your particular type of phone. In particular, occasionally a carrier may enable Visual Voicemail only for Apple iOS devices (ie, Apple iPhone) and not for Android devices.