One tesla is exactly 10,000 gauss. The tesla is the SI unit and the gauss its CGS predecessor, and the four orders of magnitude between them reflect how differently the two systems scaled magnetic quantities.
About the units
The tesla is one weber per square metre, or equivalently one newton per ampere-metre. It is named after Nikola Tesla and is a very large unit: the Earth's magnetic field is only 25 to 65 microtesla depending on latitude. The gauss, from the CGS electromagnetic system, is sized for everyday magnets — the Earth's field is about 0.25 to 0.65 gauss, and a refrigerator magnet around 50. Both units remain in active use, split by field of work: physics and medicine in tesla, geophysics and permanent magnet engineering in gauss.
The exact factor
The factor is exactly 10,000, since a tesla is one weber per square metre and a gauss is 10⁻⁴ of that. Reference values: Earth's field 0.25 to 0.65 G or 25 to 65 µT; a fridge magnet 50 G or 5 mT; a strong neodymium magnet at its surface 5,000 G or 0.5 T; a clinical MRI scanner 15,000 to 30,000 G or 1.5 to 3 T; and the strongest continuous laboratory magnets about 45 T. Neutron stars reach 10¹² T, which is where the numbers stop meaning anything intuitive.
Where you meet this conversion
Magnetic resonance imaging, where scanner strength is the headline specification and always in tesla — a 3 T machine gives better signal than a 1.5 T one at the cost of more artefacts. Permanent magnet catalogues specify surface field in gauss. Geophysical surveys and magnetometers work in nanotesla, where one nanotesla was formerly called a gamma. Loudspeaker magnets, motor design and magnetic separation equipment are specified in gauss, while the physics behind them is calculated in tesla.