SAP Basis or SAP Basic Components represents the technical layer of SAP systems on which SAP applications are developed and executed. At time of writing this article, this technical platform is named Netweaver and is common to many SAP systems (ERP, CRM, SRM, BI, …).
For SAP ERP for example, SAP Netweaver supports functional components (like Finance, Purchasing, …) by providing a set of technical functionalities to develop, execute and interface applications, maintain the data dictionary (= the database definition), organized authorizations and much more.
SAP Basis is the component that interests primarily SAP system administrators and SAP developers. While system administrators are mainly interested by the system administration functions offered by this component, developers are mainly interested by the data dictionary maintenance and the ABAP Development Workbench and its associated tools.