What the Kp index is
Kp is a 0–9 index of how disturbed Earth's magnetic field is, published in three-hour blocks. It describes the size of the auroral oval rather than the aurora above you: the higher the Kp, the farther from the pole the oval reaches. The value shown as current is NOAA's real-time estimate; the definitive index is computed later by GFZ Potsdam. Storm levels on NOAA's G-scale start at Kp 5 (G1) and end at Kp 9 (G5).
What the number means at your latitude, and why inside the oval it matters less, is a guide of its own: what the Kp index means.
Reading the chart and the table
The chart and the table above are NOAA's forecast, not ours, in the three-hour UTC blocks NOAA publishes. Blocks marked "observed" are measured, the current one is "estimated", the rest are forecast. How far ahead each of them deserves trust is a section of the guide: how far ahead a Kp forecast can be trusted.
Kp is also only one input of the forecasts on this site. The Aurora Score of a place weighs this Kp against the place's own threshold and adds cloud cover, darkness and the moon — which is why two places under the same Kp get different verdicts.
How our forecast works has the arithmetic.