We propose Slim Graph: the first programming model and framework for practical lossy graph compression that facilitates high-performance approximate graph processing, storage, and analytics. Slim Graph enables the developer to express numerous compression schemes using small and programmable compression kernels that can access and modify local parts of input graphs. Such kernels are executed in parallel by the underlying engine, isolating developers from complexities of parallel programming. Our kernels implement novel graph compression schemes that preserve numerous graph properties, for example connected components, minimum spanning trees, or graph spectra. Finally, Slim Graph uses statistical divergences and other metrics to analyze the accuracy of lossy graph compression. We illustrate both theoretically and empirically that Slim Graph accelerates numerous graph algorithms, reduces storage used by graph datasets, and ensures high accuracy of results. Slim Graph may become the common ground for developing, executing, and analyzing emerging lossy graph compression schemes.