A rope holding a still load pulls with exactly the weight, $mg$. Start lifting that load with an upward acceleration and the rope has two jobs at once: beat gravity, AND supply the $ma$ the acceleration demands. Both come out of the same cable, so $T = m(g + a)$ — read aloud, T equals m, bracket, g plus a. Here $T$ is the rope tension in newtons, $m$ the mass being lifted in kilograms, $g = 9.8\ \mathrm{m/s^2}$, and $a$ the upward acceleration in $\mathrm{m/s^2}$, taken positive when the load is gaining upward speed.

That plus sign is the whole lesson, and it is worth a sanity habit: at $a = 0$ the relation collapses to $T = mg$, which is the still-load answer, so the shape is right. It is also the lurch you feel in a lift starting upward — for a few seconds everything aboard, you included, pulls harder on whatever holds it.

Hang two masses over one free-running pulley instead and you get Atwood's machine. The out-of-balance weight $\left(m_1 - m_2\right)g$ is what drives it, but BOTH masses have to move, so it is shared over $m_1 + m_2$: $a = \dfrac{\left(m_1 - m_2\right)g}{m_1 + m_2}$. Subscripts by convention: $m_1$ is the heavier, descending mass in kilograms and $m_2$ the lighter, rising one, so the difference on top stays positive. The result is always less than $g$ — the lighter side is dead weight to haul — and that is exactly why Atwood built the thing in 1784: it dilutes free fall until a person with a pendulum clock can time it.