Assuming you mean Oracle 10g express edition, that is, a free version of the Oracle database that supports a limited amount of hardware. You are limited to 4 GB of user data (this limit is increased to 11 GB in the express version of Oracle 11g), 1 GB of RAM and 1 processor, but all this is free. Oracle competes with other free editions of other database vendors such as Microsoft. Keep in mind that Oracle offers many products with the word express in the name - Oracle Express is an old OLAP product, Oracle Application Express (APEX) is a PL / SQL web application development environment, express version 10g or 11 g Oracle database do not have data etc.
Enterprise Manager, on the other hand, is a tool that allows database administrators to control many different servers (database servers, application servers, etc.). Non-specific versions of Oracle come with a standalone version of Enterprise Manager that allows you to manage a single database. Downloading Enterprise Manager is typically a version of "grid control" that allows for more centralized management.