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We present the Next Generation Deep Extragalactic Exploratory Public (NGDEEP) Survey, a deep slitless spectroscopic and imaging Cycle 1 JWST treasury survey designed to constrain feedback mechanisms in low-mass galaxies across cosmic time. NGDEEP targets the Hubble Ultra Deep Field (HUDF) with NIRISS slitless spectroscopy (f~1.2e-18 erg/s/cm^2, 5sigma) to measure metallicities and star-formation rates (SFRs) for low-mass galaxies through the peak of the cosmic SFR density (0.5<z<4). In parallel, NGDEEP targets the HUDF-Par2 parallel field with NIRCam (m=30.6-30.9, 5sigma) to discover galaxies to z>12, constraining the slope of the faint-end of the rest-ultraviolet luminosity function. NGDEEP overlaps with the deepest HST ACS optical imaging in the sky: F435W in the HUDF (m=29.6), and F814W in HUDF-Par2 (m=30), making this a premier HST+JWST Deep Field. As a treasury survey, NGDEEP data is public immediately, and we will rapidly release data products and catalogs in the spirit of previous deep field initiatives. In this paper we present the NGDEEP survey design, summarize the science goals, and detail plans for the public release of NGDEEP reduced data products.