The Time Dependent Schrodinger Equation has the form
$$i\hbar\frac{\partial}{\partial{t}}\Psi=-\frac{\hbar^2}{2m}\left(\nabla^2+V\right)\Psi$$
and the Time Independent Schrodinger Equation has the form
$$E\Psi=-\frac{\hbar^2}{2m}\left(\nabla^2+V\right)\Psi$$
I know that $V$ is the potential operator and is a function of the position, and $Psi$ is the wavefunction. I also know that $\hbar$ is a constant, and $m$ is the mass of the particle.
What does $\partial$ mean, and what does $\nabla$ mean?
First of all if you know what $y'(x)$, means then you can think of
$$\frac{\partial{y}}{\partial{x}}$$ as just another way of writing $y'(x)$
and you can use other variables, so you could have
$$\frac{\partial{A}}{\partial{B}}$$
which is the same as $A'(B)$
So if for example
$A=B^2$ then the answer to
$$\frac{\partial{A}}{\partial{B}}$$
is
$$\frac{\partial{A}}{\partial{B}}=\frac{\partial(B^2}{\partial{B}}=\left(B^2\right)'=2B$$
If on the other hand I $\Lambda=\cos(\gamma)$
then the answer to
$$\frac{\partial\Lambda}{\partial\gamma}$$
is
$$\frac{\partial\Lambda}{\partial\gamma}=\frac{\partial\cos(\gamma)}{\partial\gamma}=\left(\cos(\gamma)\right)'=-\sin(\gamma)$$
$$\frac{\partial}{\partial{N}}w=\frac{\partial{w}}{\partial{N}}$$
and if for instance $w=\ln(N)$
then the answer to
$$\frac{\partial}{\partial{N}}w$$
is
$$\frac{\partial}{\partial{N}}w=\frac{\partial{w}}{\partial{N}}=\frac{\partial\ln(N)}{\partial{N}}=\frac{1}{N}$$
Now if you have seen the laplacian in the form $\nabla^2$ then it can help to see the equation
$$\nabla^2\Psi=\left(\frac{\partial\frac{\partial{\Psi}}{\partial{x}}}{\partial{x}}+\frac{\partial\frac{\partial{\Psi}}{\partial{y}}}{\partial{y}}+\frac{\partial\frac{\partial{\Psi}}{\partial{z}}}{\partial{z}}\right)$$
so the Time Dependent Schrodinger Equation is
$$i\hbar\frac{\partial\Psi}{\partial{t}}=-\frac{\hbar^2}{2m}\left(\frac{\partial\frac{\partial{\Psi}}{\partial{x}}}{\partial{x}}+\frac{\partial\frac{\partial{\Psi}}{\partial{y}}}{\partial{y}}+\frac{\partial\frac{\partial{\Psi}}{\partial{z}}}{\partial{z}}\right)+V\Psi$$
while the Time Independent Schrodinger Equation is
$$E\Psi=-\frac{\hbar^2}{2m}\left(\frac{\partial\frac{\partial{\Psi}}{\partial{x}}}{\partial{x}}+\frac{\partial\frac{\partial{\Psi}}{\partial{y}}}{\partial{y}}+\frac{\partial\frac{\partial{\Psi}}{\partial{z}}}{\partial{z}}\right)+V\Psi$$
