Angle of Acceptance →
The angle of incident of light on the core for which the angle of incidence on the corecladding interface equals the critical angle is known as the "Angle of Acceptance".

Transmission of light when the incident angle is less than the acceptance angle 
For $\theta_{i}>\theta_{0}$, the angle of incidence on the corecladding interface is less than the critical angle due to which part of incident light is transmitted as shown in the figure below

Transmission of light when the incident angle is greater than the acceptance angle 
Hence the light will be transmitted through the fibre for angles of incidence on the core which are less than the acceptance angle.
The acceptance angle is the maximum angle of incidence on the core for which total internal reflection takes place inside the core.
The light entering the core in a cone of semivertical angle $\theta_{0}$ is transmitted in the core through total internal reflections. This cone is known as the acceptance cone.
Numerical Aperture→
"The sine of the angle of acceptance of the optical fibre is known as the numerical aperture of optical fibre."
The numerical aperture determines the lightgathering ability of the fibre. It measures the amount of light that can be accepted by a fibre. The numerical aperture depends upon the refractive index of the core and cladding material and does not depend on the physical dimension of the fibre. It is a dimensionless quantity that is less than unity. The value of the numerical aperture range from $0.13$ to $0.15$. A large numerical aperture implies that a fibre accepts a large amount of light from the source. It varies due to variations of refractive index in the core and it has zero value after the corecladding boundary. The number of propagation modes to multimode gradedindex fibre depends upon the parameter of numerical aperture and hence upon the relative refractive index difference $\Delta n$
Derivation of Angle of Acceptance and Numerical Aperture →
Let us consider, stepindex optical fibre into which light is launched from one end at angle $\theta_{i}$. Let the refractive index of core and cladding is $n_{1}$ and $n_{2}$.

Transmission of light when the incident angle is equal to the acceptance angle

Let us consider that light rays enter the fibre at an angle $\theta_{i}$ to the axis of the fibre refracted at an angle $\theta_{r}$ and strike the corecladding interface at an angle $\phi$.
When ray incident at point $A$ on the core then According to Snell's law
$\frac{sin \theta_{0}}{sin \theta_{r}}= \frac{n_{1}}{n_{0}} $
Where $n_{0}$ → refractive index of air and vacuum
$sin\theta_{0}=\frac{n_{1}}{n_{0}} sin \theta_{r} \qquad(1)$
Now the refracted ray incident at point $B$ at the interface of core and cladding. So for critical angle condition
$n_{1}\: sin\phi=n_{2} \: sin90^{0} $
$n_{1}\: sin(90\theta_{r})=n_{2} \: sin90^{0} \qquad (\because \phi=90\theta_{r})$
$n_{1}\: cos \theta_{r}=n_{2}$
$cos\theta_{r}=\frac{n_{2}}{n_{1}} \qquad(2)$
$sin\theta_{r}=\sqrt{1cos^{2}\theta_{r}}$
Now substitue the value of $cos\theta_{r}$ from equation$(2)$ to above equation then we get
$sin\theta_{r}=\sqrt{1 \left ( \frac{n_{2}}{n_{1}} \right)^{2}}$
$sin\theta_{r}=\frac{1}{n_{1}}\sqrt{n_{1}^{2} n_{2}^{2}}$
Now substitue the value of $sin\theta_{r}$ in equation$(1)$ then we get
$sin\theta_{0}=\frac{n_{1}}{n_{0}}\frac{1}{n_{1}}\sqrt{n_{1}^{2} n_{2}^{2}}$
$sin\theta_{0}=\frac{\sqrt{n_{1}^{2} n_{2}^{2}}}{n_{0}}\qquad(3)$

This $sin\theta_{0}$ is known as Numerical Aperture. i.e.
$N.A.=\frac{\sqrt{n_{1}^{2} n_{2}^{2}}}{n_{0}}$

If fibre is in the air then $n_{0}=1$ so the above equation can be written as
$N.A.=\sqrt{n_{1}^{2} n_{2}^{2}}$

The equation $(3)$ also can be written as
$\theta_{0}=sin^{1}\frac{\sqrt{n_{1}^{2} n_{2}^{2}}}{n_{0}}$

The angle $\theta_{0}$ is known as the Angle of Acceptance.
The light is transmitted through the fibre when
$\theta_{i} < \theta_{0}$
i.e. $sin \theta_{i} < sin \theta_{0}$
The light will be transmitted through the fibre with multiple total internal reflections when the above condition is satisfied.