One neper is 8.6858896381 decibels. Both are logarithmic ratio units rather than absolute quantities, and the whole difference between them is which logarithm they use: the neper takes the natural log, the decibel takes base ten.
About the units
A decibel expresses a ratio between two quantities on a logarithmic scale, and it is a tenth of a bel — a unit named after Alexander Graham Bell and devised at Bell Telephone Laboratories in the 1920s to describe signal loss along telephone lines. The bel turned out to be inconveniently large, so the decibel became the working unit. The neper does the same job using natural logarithms and is named after John Napier, who introduced logarithms in 1614. It is preferred in theoretical work on transmission lines and control systems, where natural logarithms fall out of the differential equations directly and converting to base ten just adds a constant.
The exact factor
The exact relationship is 1 Np = 20 ÷ ln(10) dB = 8.685889638065035 dB for field quantities such as voltage, current or sound pressure. For power quantities the constant is half that, 10 ÷ ln(10) = 4.3429 dB per neper, because power goes as the square of the field. That factor-of-two distinction is the single most important thing to get right in either unit: a doubling of sound pressure is 6.02 dB, while a doubling of sound power is 3.01 dB, and both are correct statements about different quantities. Anyone who has seen both 10log and 20log in the same document has met this.
Where you meet this conversion
The decibel is everywhere: sound levels, audio equipment gain, radio signal strength, optical fibre attenuation and antenna gain. Because it is a ratio it needs a reference to become absolute, which is what the suffixes mean — dB SPL is referenced to 20 micropascals of sound pressure, dBm to one milliwatt, dBV to one volt. The neper appears mainly in transmission line theory, where attenuation constants are naturally expressed in nepers per metre, and in control engineering. Anyone reading both an academic paper and a datasheet on the same amplifier may meet both units for the same measurement.