Q:

What is meant by local repository and remote repository in Maven speak?

I am a newbie to Maven . I am reading up Maven - The complete reference and came across this line : 

Maven assumes that the parent POM is available from the local repository, or available
  in the parent directory (../pom.xml) of the current project. If neither location is valid
  this default behavior may be overridden via the relativePath element.

What exactly is meant by local and remote repository for a maven installation and a project

A:

A local repository is a local directory structure that caches artifacts downloaded from remote repositories, or those that are manually installed (eg from the command line option).
A remote repository is a web service (defined by a URL) that contains versioned artifacts. This might be as simple as an Apache server, or a full-blown Maven repository, such as Artifactory, that allows uploading, permissions based on a user directory, etc. 
http://maven.apache.org/guides/introduction/introduction-to-repositories.html

