Two Line Elements or TLEs, often referred to as Keplerian elements or keps in the amateur community, are the inputs to the SGP4 standard mathematical model of spacecraft orbits used by most amateur tracking programs. They specify the size and shape of the orbit, and how the orbit is oriented with respect to the Earth at a particular moment in time. With the TLEs, and your station location, you can compute when the satellite will be in view and where to point your antennas. See the tutorial on Keplerian elements and the explanation of the formats used in these files. Details of the mathematics behind these elements may be found in Spacetrack Report No. 3 and Revisiting Spacetrack Report #3.
Weekly updates are completely adequate for most amateur satellites. A significant exception is the International Space Station. Because of its low orbit, it experiences significant aerodynamic drag which over a few days introduces noticeable errors, particularly for the 2395 MHz HamTV transmissions that require narrow beamwidth dish antennas for good reception. Additionally, its orbit is periodically raised by thrusters. These combined factors require frequent updates to ensure accurate predictions. To satisfy this requirement, AMSAT has partnered with the Sci-Tech Amateur Radio Society to generate a daily bulletin.
When a new satellite is launched there is some confusion regarding TLEs. Look to amsat-bb, Twitter and Facebook for the latest information.
Bulletin format elements for all satellites of interest to radio amateurs. This file is updated at least every 24 hours.
Bare elements are derived from the bulletin, but contain no header or footer. These data are used as input to the AMSAT Pass Predictions page.
Historical bulletins are available in The Keps Archives. Historical elements are available to support AMSAT’s Fox series satellites and the FoxTLM software. You may also find bulletins dating back to October 1993.
You can find lots of TLEs and related information on these other sites, including NOAA weather, Iridium, and all other unclassified satellites not strictly of interest to the amateur radio community:
CelesTrak by T.S. Kelso. TLEs for selected spacecraft, updated 4-6 times a day.
Space-Track United States Space Force site for TLE’s going back to Sputnik 1. Requires a free account for access. Current satellites updated 4-6 times a day.