What does 9.8N/kg mean?

Force is equal to the rate of change of momentum in accordance with Newton's second law of motion. Force is defined as mass times acceleration for a constant mass.
$\mathrm{Force}=\mathrm{mass}\times \mathrm{acceletration}\mathrm{acceleration}=\frac{\mathrm{Force}}{\mathrm{mass}}$
Force per mass means Newton per kg. $9.8N/\mathrm{kg}$ means $9.8N$ earth's gravitational field strength is required to pull $1\mathrm{kg}$ mass towards the earth's center.