One sievert is exactly 100 rem. Both measure equivalent dose, which is an estimate of biological harm — not an amount of energy, and not the same thing as the gray, even though all three have the dimensions of joules per kilogram.
About the units
The sievert is the SI unit of equivalent dose. It starts from the absorbed dose in gray and multiplies by a radiation weighting factor: 1 for X-rays, gamma and beta radiation, 2 for protons, and up to 20 for alpha particles and neutrons, because those deposit their energy in dense tracks that damage DNA far more effectively. One gray of alpha radiation is therefore 20 sievert of equivalent dose. Effective dose goes further and applies tissue weighting factors as well, since bone marrow and gonads are more radiosensitive than skin or bone. The rem is the older unit, exactly 0.01 Sv, from Roentgen Equivalent Man.
The exact factor
The factor is exactly 100. The relationship to the gray is the one that matters and cannot be reduced to a factor: sieverts equal grays multiplied by weighting factors that depend on the radiation and the tissue, so no conversion between gray and sievert exists in general. This is precisely why the two must never share a conversion table — a fact this project learned the hard way, since gray, sievert and becquerel once sat in the same category here and were silently converted into one another.
Where you meet this conversion
Radiation protection, dose limits and risk assessment. Occupational limits are set in millisieverts per year — 20 mSv in any single year for radiation workers in the EU under Directive 2013/59/Euratom, 1 mSv for the general public above background. The directive allows one narrow exception: a member state may permit up to 50 mSv in a single year provided the average across five consecutive years stays at 20. Nuclear medicine, aviation crew exposure and radon risk are all quantified in sieverts. The rem persists in American regulation and older literature, where an annual occupational limit of 5 rem is the same as 50 mSv. Anyone comparing American and European radiation regulations converts between them constantly.