Hi,
The App is Messages.
It can have an iMessages account is use.
This distinction is becoming more important as time goes on.
In Yosemite the Messages App can start a Video (or Audio and also Screen Sharing) two different ways. (it's related bear with me).
As the iChat app the AIM and Jabber accounts have been able to Video chat within the app since iChat 2 for AIM accounts and iChat 3 for Jabber when it was added. (Screen Sharing is added at iChat 4)
This needs the Video Menu > Enable the three items for Video, Audio and Screen Sharing.
In Yosemite the iMessages account can start a video from the Main window when a text Chat is going on using the Details option to right.
To start an AIM Video chat (same for Jabber and Bonjour) you need the Buddy list to be displayed, Highlight the Buddy and then use either the icon at the bottom of the Buddy List or the Buddies menu.
To Start an iMessages Video chat you need to first set up FaceTime with the same Apple ID account (it can be done with a different ID but this can cause confusing at the other end about who is calling).
Then you use the Details option having started a Text (iMessages) chat with them.
This in turn Invokes the FaceTime App and 'Calls' the ID your are iMessaging to. If you are using a different ID in FaceTime they see that and not the one in the iMessages account.
Screen Sharing
Screen Sharing is similar in that it (iMessages account) invokes an outside app to do the Screen Sharing.
You first need to go to System Preferences > Sharing and enable the Screen Sharing item.
Then in Messages in the Main window during a iMessages Chat/conversation with someone you need to use the Details option top right and select the Linked Squares for the Screen Sharing.
This invokes the Screen Sharing App (System/Library/Core Services/Applications) and connects the two computers with and Audio chat along side.

For the iChat type accounts (AIM, Jabber and Bonjour) you use the Buddy list like the Video chats.

NOTES.
If you have other Macs on your LAN and they appear in the Finder's Side bar as Shares you can select them and use the button in the Finder Window to Screen Share with those Macs.

This too invokes the Screen Sharing app.
Another way is to use the Go Menu in the Finder > Go to Server and enter VNC:// and the IP address instead of AFP or SMB (Or Browse instead)
In the System Preferences > Sharing you will also find Remote Management.
This cannot be on the same time as Screen Sharing. However in use from the Finder it allows the Screen Sharing described above.
All of these use the Client end of Apple Remote Desktop (ARD).
However the Finder methods described do not come wither Audio Chat described for the iMessages or AIM/Jabber/Bonjour way.
Also if you try to use the Details method with an AIM or Jabber of Bonjour Buddy it will not work as it tries to do this the iMessages way.
Basically "icons" and "Messages" is not enough to say where you are in the app to pin point this further.

9:49 pm Saturday; March 7, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad