What means do airline ground staff have to converse with pilots in flight?
There are numerous methods for air-to-ground communications.  I may be missing a few, but off the top of my head, here are the ones I can remember:


normal VHF comms
Operational staff at the airport maintain an operations frequency that you can use just as you would would ATC.  The limiting factor here is the requirement of line-of-sight with the ground antenna and this limits range.  We could normally raise ops about 30-40 minutes out, at altitude, if someone was actually listening.  This is similar to a GA pilot calling unicom to arrange ground services.  This method is also used for maintenance at some stations and for talking to fuelers and deicing operators.
SELCAL
This is a selective calling protocol that operates over HF and VHF frequencies.  Each participating airplane and ground station (typically a dispatcher or ATC) has a code and if you know the code you want to communicate with, you can establish two-way communications with them.  To use this in the air required tuning a specific frequency and using a hand-mic with a keypad to dial the ground station code.  On the ground I'm not sure the exact method they use (phone patch?) but when they called us we'd get an aural "SELCAL" announced by the airplane along with a light to alert us.  
AIRINC phone patch
This service provides VHF-to-PSTN patches and allows you to call your dispatchers actual phone, medlink, or anyone with a phone you feel like calling.  This worked by calling an operator over the normal radios and having them dial a number and connecting us together.
Satellite phone
Works just like any other satellite phone and use the Iridium network.
ACARS
This works just like text messaging.  Select your recipient, type a message and hit send.  You get alerts when messages arrive.  The interface I'm familiar with in the airplane is a program in the FMS with limited pre-set recipients you could contact (dispatch and perhaps a few others).  The ACARS software can also be setup to send specific messages to specific stations, e.g. an "In-Range" message which is routed to operations at your destination.  From the ground side, this is usually baked into the dispatch flight-following software.  Our dispatchers could right click on an airplane, get a menu, pick "text message" and type a message that would be sent to that airplane.