What is the meaning of $\exp(\,\cdot\,)$? What is the meaning of the notation $\exp(\text{expression})$ ?
I think that it's something of the form $a^\text{expression}$ but does it mean that the base $a=e$ or can it be any base?
 A: The notation $\exp(x)$ means $e^x$. See here on Wikipedia for example.
A: To answer your question, no - it isn't defined for an arbitrary base $a$, only $e$ which is the value you hit when you try to take the limit as $n$ goes to $0$ to the following expression (as Mr. Bernoulli historically encountered the elusive $e$):
$$\lim_{n\rightarrow0}\left(1+\frac{1}{n} \right)^n = L$$
Where $L = e$ or $2.71828182845\ldots$ Finally:
$$\exp(x)=e^x$$
You can see why it would be prudent to use the left side when you have a hefty exponent which is quite natural for the uses of $e$ (think Gaussian function, normal distribution, etc).
Usually, for other bases, especially in computing/programming, you'd use a more generic function such as $\operatorname{pow}(a, b)$ which basically returns $a^b$.
Hope it helps.
A: $\exp(x)=e^x$
Where $e$ is Euler's Number (approx. $2.718281828$).
This notation is common when $x$ is complicated and messy when put in the form $e^x$ (like large fractions or polynomials, for instance).  Also, $\exp$ is the name of the $e^x$ function in many programming languages.
$e$ is a very important constant that appears in many fields of mathematics.  The inverse of the $\exp(x)$ function is the natural logarithm (often written as $\log(x)$, $\ln(x)$ or $\log_e(x)$).
A: $\exp(x)$ and $e^x$ refer to the same function, but I'm not sure I would say $\exp(x)$ means $e^x$.
Consider the expression $b^x$, where $b$ is real and $x$ is irrational.  How would you define it?
I would say, $b^x = \exp(\ln(b^x)) = \exp(x \ln(b))$, where $\exp$ means the antilogarithm.  The expression $e^x$ is, of course, a special case of $b^x$.  If your book is taking this approach, you might find the theorem $\exp(x) = e^x$ proved later on.
