I am using TeamCity (version 9.1.5 if that matters) and I am trying to figure out how to create a trigger that deploys the project to a server. Or maybe there is a way to deploy a project to a server without using a trigger on TeamCity.
It's a very broad question, but I will share the approaches I have used in a couple of scenarios:
1) To deploy when a code checkin is performed, I have setup a build configuration that does the deployment, added the build configuration that does the compiling & packaging as a snapshot and artefact dependency which is then triggered with a Finish Build Trigger https://confluence.jetbrains.com/display/TCD9/Configuring+Finish+Build+Trigger
2) To deploy at a given time of the day but only when new code has been checked in, I have setup a build configuration as above but triggered with a Schedule Trigger https://confluence.jetbrains.com/display/TCD9/Configuring+Schedule+Triggers ensuring to select the dependent build in the Build Changes section.
With regards to how to perform the deployment there are many options, I have used WebDeploy for ASP.Net applications and MSI packages executed by Remote Powershell scripts for Windows Services, but other options are also available depending on the technology you have.
JetBrains provide an end to end example for ASP.Net in their on-line documentation, search for "Continuous Delivery to Windows Azure Web Sites (or IIS)"