can someone fully explain what a derivative and an integral is?

Let us first understand why derivative and integration are used and what do they signify!

Derivative can be used to determine the rate of a quantity with respect to time or space, slope and gradient and even maximising and minimising a particular function while integration is used to determine the total or summation over particular range or area under a curve within a specified domain.

Basic example:
Consider a bike moving with a varying velocity with respect to time given by $$v(t)=2t$$
Know these two concepts: Slope of velocity-time curve gives acceleration at that point and area under velocity-time curve gives the displacement of the object during a specified range.

Standard Formulas:$$\frac{d\;(t^n)}{dt}=nt^{n-1}\;\;\;\;\;and \;\;\;\;\;\;\;\;\int t^n dt= \frac{t^{n+1}}{n+1}+c, n\ne1$$ where c is the arbitrary constant of integration.
There are lots of standard formulas and you can find them on google. And slowly from basics you will learn methods of finding differentiation and integration of other complex functions which is the most interesting thing in this subject. 

So, let us find the acceleration of the object at any time 't'.
$$a=\frac{dv}{dt}=2(1\cdot t^{1-1})=2\;m/s^2$$

Let us find the displacement of the bike from time $t=0$ to $t=1$ seconds
$$s=\int\limits_0^1 v dt=2\left[\frac{t^2}{2}\right]_0^1=[t^2]_0^1$$
Note that arbitrary constant is not used because we have specified range here.
Now to solve it plug upper limit wherever t comes and subtract it with plugging lower limit wherever t comes in the expression.
$$s=1^2-0^2=1\;m$$


This is one application of derivatives and integrals. There are lots of applications, infact whole calculus is very beautiful. You will learn lots of real life applications in this subject.
Good Luck!