What are SAP and ABAP? I searched and got a bunch of different acronyms that don't quite make sense.
What are they primarily used for?
I have worked with SAP since 1998. SAP is a type of software called ERP (Enterprise Resource Planning) that large companies use to manage their day to day affairs. On the macro, the software can be split into two categories: Technical and Functional
Let's go Technical first, as it answers the "What is ABAP" part of your question.
There are two technical "stacks" within the SAP software, the first is the ABAP stack which is inclusive of all the original technology that SAP was. ABAP is the proprietary coding language for SAP to develop RICEFW objects (Reports, Interfaces, Conversions, Extensions, Forms and Workflows) within the ABAP stack.
The ABAP stack is traditionally navigated via Transaction Codes (T-Codes) to take you to different screens within the SAP Environment. From a technical perspective, you will do all of your performance and tuning of the WORK PROCESSES in the SAP system here, as well as configuring all of the system RFCs, building user profiles and also doing the necessary interfacing between the OS (usually Windows or HPUX) and the Oracle Database (currently Enterprise 11g).
The JAVA stack controls the "Netweaver" aspect of SAP which encapsulates SAP's ability to be accessed via the Internet via SAP Portal and it's ability to interface with other SAP and non-SAP legacy systems via Process Integration (PI).
SAP also has extensive capabilities in the Business Intelligence Field (BI) by accessing information stored within the Business Warehouse (BW). Currently, there is a new technology called HANA 1.0 that compresses the time to run reports against these repositories.
There are two primarily technologists that run ALL of these functions, they are called SAP Basis (Netweaver) Administrators and ABAP Developers.
SAP has specific pre-populated functional packages for different business areas. For example, Exxon runs the "IS Oil & Gas" package while Bank of America runs the "Banking" package, while further still Lockheed Martin runs the "Aerospace & Defense" package. These packages were developed over time by the amalgamation of intelligent functional customizations that could be intelligently ported to the system via inclusion in dot releases.
However, there are some vanilla functional modules that almost all entities run, regardless of their specific industry:
and finally the biggie: