One newton is exactly 100,000 dynes. The dyne is the force unit of the centimetre-gram-second system, and it survives in a few corners of physics where the quantities involved are genuinely tiny.
About the units
The dyne is the force that accelerates one gram at one centimetre per second squared. It belongs to the CGS system, which competed with the metre-kilogram-second system through the nineteenth and early twentieth centuries and lost — but not everywhere. CGS units persist in astronomy, in some areas of electromagnetism and in surface science, largely because the historical literature is written in them and rebasing decades of published values is unattractive. A dyne is a very small force: roughly the weight of a milligram, or the force needed to lift a large grain of sand.
The exact factor
The derivation is straightforward: a gram is 10⁻³ kg and a centimetre per second squared is 10⁻² m/s², so a dyne is 10⁻⁵ N, and a newton is 10⁵ dynes. Exactly. The related CGS quantity most often met is surface tension, measured in dynes per centimetre, which converts to millinewtons per metre one for one — water at 20 °C has a surface tension of 72.8 dyn/cm, which is 72.8 mN/m. That numerical identity is a rare convenience in a system otherwise full of awkward factors.
Where you meet this conversion
Surface science and materials, where surface tension and surface energy are still quoted in dynes per centimetre. Adhesion testing of plastics uses dyne pens, which mark a surface with inks of known surface tension to establish whether it will accept printing or bonding. Astrophysics uses CGS throughout for pressure, energy and magnetic fields. In seismology and older mechanics literature the dyne appears in force balances. Outside those fields it has been entirely displaced by the newton.