We present a search for spatial extension in high-latitude (vertical bar b vertical bar > 5 degrees) sources in recent Fermi point source catalogs. The result is the Fermi High-Latitude Extended Sources Catalog, which provides source extensions (or upper limits thereof) and likelihood profiles for a suite of tested source morphologies. We find 24. extended sources, 19 of which were not previously characterized as extended. These include sources that are potentially associated with supernova remnants and star-forming regions. We also found extended.-ray emission in the vicinity of the Cen. A radio lobes and-at GeV energies for the first time-spatially coincident with the radio emission of the SNR CTA 1, as well as from the Crab Nebula. We also searched for halos around active galactic nuclei, which are predicted from electromagnetic cascades induced by the e(+)e(-) pairs that are deflected in intergalactic magnetic fields. These pairs are produced when gamma-rays interact with background radiation fields. We do not find evidence for extension in individual sources or in stacked source samples. This enables us to place limits on the flux of the extended source components, which are then used to constrain the intergalactic magnetic field to be stronger than 3 x 10(-16) G for a coherence length lambda greater than or similar to 10 kpc, even when conservative assumptions on the source duty cycle are made. This improves previous limits by several orders of magnitude.