Half life (mathematically T_(1/2)) is how long it takes for half of the atoms in a substance to radioactively decay.
If you want to know the maths behind their relationship,
N = N_0e^(-lambdat) applies to radioactive substances, where
N is the number of radioactive atoms at time t
N_0 is the number of radioactive atoms at the beginning of the process, when t = 0
e is Euler's constant, approx 2.71828
t, as mentioned, is time, and
lamda is the decay rate, which is a constant value for each isotope. It can also be thought of as the probability for an atom to decay in a unit time.
When t = T_(1/2), then half the initial atoms have decayed, which means that N = N_0/2.
Substituting this into the equation,
N_0/2 = N_0e^(-lambdaT_(1/2))
1/2 = e^(-lambdaT_(1/2))
Taking natural logs and rearranging from there,
ln(1/2) = -lambdaT_(1/2)
ln1 - ln2 = -ln2 = -lambdaT_(1/2)
ln2 = lambdaT_(1/2)
T_(1/2) = ln2/lambda = 0.693/lambda
which is mathematically how the rate of radioactive decay is related to half life.
From this equation, we can see that if decay rate (lambda) increases, T_(1/2) will get shorter, and if decay slows down, half life will increase.
They are inversely proportional.
