How can I move customer community from one Dev org to another Dev org. I use maven mate and eclipse to move meta data.
You can move community using Eclipse IDE. You can follow the steps given here.
The only difference here is instead of Change set you will use Eclipse IDE. In the IDE Select the Network component type, choose your community, and then add it. Select all the dependency as well and then deploy it to another org. There are some prerequisite steps as well which you need to follow as well.
If it is a Lightning Community you can follow steps here : Ways to Deploy Lightning Communities
You need Data Loader:
The Force.com platform offers you a development environment which you can use to create your own custom applications. Frequently, you will use the Force.com platform to create applications that operate on data that you already have in some form - in comma-separated variable files, spreadsheets or other relational databases.
The Data Loader is an easy to use graphical tool that helps you to get your data into Salesforce objects. The Data Loader can also be used to extract data from database objects into any of the destinations mentioned above. You can even use the Data Loader to perform bulk deletions by exporting the ID fields for the data you wish to delete and using that source to specify deletions through the Data Loader.
The Data Loader requires the use of the Force.com API. If your Salesforce edition allows the use of the API (Enterprise, Unlimited and Developer Editions), you can download the Data Loader from the Setup menu, under Administer heading - Data Management.