The word "integral" is used in two completely different senses. The first, called definite integral, has a simple geometric (or physical) interpretation, the second, called indefinite integral, is accessible only to people having the notion of "derivative of a function of one variable" in their repertoire. It is true that in the one-dimensional case there is a connection between the two notions. This connection is called the fundamental theorem of calculus.
(a) The definite integral: Given some sort of "intensity" $f(x)$ at each point $x$ of some domain $B$ (an interval, a sphere, a cube in ${\mathbb R}^n$, etc.), where $f(x)$ varies with $x$, one can ask for the "total effect" an agent of this intensity could have. This total effect is the integral of $f$ over $B$ and is denoted by $\int_B f(x){\rm d}(x)$ (or similar). From the geometric intuition behind it this quantity is a limit of Riemann sums, viz. $\int_B f(x){\rm d}(x)\ =\ \lim_{\ldots} \sum_k f(\xi_k)\ \mu(B_k)\ ,$ where the $B_k$ form a disjoint partition of $B$ into very small subdomains and $\mu$ denotes the natural geometric measure (length, surface area, $n$-dimensional volume) in the situation at hand.
(b) The indefinite integral: Given a function $t\mapsto f(t)$ on some interval $I\subset{\mathbb R}$ one may ask: Is this function the derivative of some other function $F(\cdot)$? The answer is yes, and in fact there is an infinite set of such functions $F(\cdot)$, whereby any two of them differ by a constant on $I$. This set of functions is called the indefinite integral of $f$ on $I$ and is denoted by $\int f(t)\ dt\ .$
(c) The fundamental theorem of calculus: Given two points $a$, $b\in I$ the difference $F(b)-F(a)$ has the same value for all functions $F\in\int f(t)\ dt$ and may as well be denoted by $\int_a^b f(t)\ dt\ .$ Now comes the theorem (and this is the real wonder, not the fact that taking the derivative of the antiderivative of $f$ gives back $f$): When $a then $\int_{[a,b]} f(t)\ {\rm d}t = \int_a^b f(t)\ dt\ .$ Here on the left side we have a limit of Riemann sums, and on the right side a difference of $F$-values.