Unlike Jenkins, TeamCity is a proprietary product that is free to use only for open source projects. It is created and managed by JetBrains. This is the same company that also created IntelliJ IDEA, which is the IDE that Android Studio is based on. TeamCity supports Android builds with Gradle out of the box.
If you do not have a running TeamCity installation yet, download it from the JetBrains website (https://www.jetbrains.com/teamcity) and follow the steps.
To get started with building Android apps and libraries with TeamCity, you need to make sure the JDK, the Android SDK, and the Android build tools are installed on your build server. You can find the instructions for this in the Jenkins section. You also need to add ANDROID_HOME to the environment variables of the machine, and point it to the Android SDK directory.
Unlike Jenkins, TeamCity does not require any plugins to trigger Gradle builds, as TeamCity has built-in support for running Gradle.