Root Browser Lite is a file manager for both rooted and non-rooted Android devices, wih a beautiful interface and lots of useful features. For rooted phones, it gives you access to all of Android's file system. You can select multiple files and batch copy, move, zip, delete, tar. An sqlite database viewer is also included
AndExplorer is a file manager for Android-powered devices. It allows browsing files and folders stored on device and sdcard. It can sort by name, size and date. Rename, delete, create folder features are available. It is able to uncompress ZIP, GZIP and TAR files too. It can be called from another application through Intent facilities
X-plore for Android is a file manager for Android devices, with a wide range of functions on the phone's file system.
It allows to explore content of Linux file system (read-only on non-rooted phones), as well as memory card, where files may be read, deleted, renamed, moved, zipped, etc