What is the meaning of integral of negative magnetic field (nT) over time?
Disturbance storm time index is a measure of the weakened horizontal component of Earth magnetic field during great magnetic disturbances. The depression is often flanked by peaks, that initiates and ends the storm. Dst is published by Kyoto University and more information about the methods and references are available on the web page. 


  The Dst index represents the axially symmetric disturbance magnetic field at the dipole equator on the Earth's surface. Major disturbances in Dst are negative, namely decreases in the geomagnetic field. These field decreases are produced mainly by the equatorial current system in the magnetosphere, usually referred to as the ring current. The neutral sheet current flowing across the magnetospheric tail makes a small contribution to the field decreases near the Earth. Positive variations in Dst are mostly caused by the compression of the magnetosphere from solar wind pressure increases. 


Sugiura and Kamei, IAGA Bulletin No 40 (1991)

Measures are presented as the deviation from the average magnetic field strength, measured in the SI unit Tesla $T$ at time $t$. In this context, the magnetic field is very weak, so the nano prefix $n$ is usually used. Magnetic field strength can be derived from weight (in $kg$), electric charge (in Coulomb $C$) and time in second ($s$) as:

$T =\frac{kg}{C \times s}$

if you integrate with time (in seconds, $s$) you'll get for each unit: 

$T \times s =\frac{kg}{C}$. 

The relation of electric charge and weight is used as a measure of Radiation exposure, often given in röntgen ($R$) and related as:  

$1 \frac{C}{kg} = 3876 R$

Technically, your integral will, therefore, be a measure of increased radiation exposure, kinetic energy released per unit mass, for the time you integrate over. You could convert it to röntgen ($2.58×10^{−4} C/kg = R$) but note that the calculated product describes the deviation, not the absolute ionization.