What is the meaning of the DFT?
As you noted, the discrete Fourier transform (DFT) maps a length-$N$ sequence to its (also length-$N$) frequency-domain equivalent. The time-domain signal $s[n]$ is transformed into a frequency-domain signal $S[k]$, such that the following two relations hold true:

$$
S[k] = \sum_{n=0}^{N-1} s[n] e^{\frac{-j2\pi k n}{N}} \text{ (the DFT)}
$$
$$
s[n] = \sum_{k=0}^{N-1} S[k] e^{\frac{j2\pi k n}{N}} \text{ (the inverse DFT)}
$$
(ignoring scale factors that can be placed in many different places depending upon the preference of the author; one way to make them "symmetric" is to place a factor of $\frac{1}{\sqrt{N}}$ in front of each equation: the unitary DFT).

There are a number of intuitive interpretations for the action implied by these equations. Here's the way I usually think about them:


Forward DFT: Each DFT output value $S[k]$ is a complex value that represents the amplitude and phase of $s[n]$'s content at frequency $\frac{2\pi k}{N}$. The multiplication by the exponential function effectively shifts $s[n]$ down in frequency by $\frac{2\pi k}{N}$. The sum over $N$ time samples can then be thought of as applying a decimating lowpass filter. So, effectively, the $N$ outputs of the DFT represent the results of applying a bank of equally-spaced filters across $s[n]$'s frequency band, thus measuring "how much" energy is present in various frequency bands in the input signal.
Inverse DFT: The DFT output values $S[k]$ are used to weight a number of complex exponential functions in order to resynthesize the original time-domain signal $s[n]$. The frequency associated with $S[k]$ is the additive inverse of the frequency used during the forward DFT. This view clearly shows that the DFT can be thought of as a means to decompose an arbitary finite-length time-domain signal $s[n]$ into a set of weighted orthogonal complex exponential functions (or "complex sinusoids").


Now, extending this concept a bit: when you sample a continuous-time signal $s(t)$ at discrete time instants, you get a (potentially infinite-length) discrete-time signal $s[n]$. Applying the Fourier transform to such a discrete-time signal results in the discrete time Fourier transform (DTFT), which is continuous in frequency and, like the DFT, periodic in frequency with period $2\pi$. Assuming that a large-enough sample rate was used during the conversion to discrete-time, the result of the DTFT will look a lot like the Fourier transform of the original signal $s(t)$.The DFT is only applicable to finite-length signals, so in order to apply it, one must truncate the discrete-time signal $s[n]$ to some finite length $N$, resulting in a potentially-shorter signal $s_N[n]$. 


For this special case of a finite-length discrete-time signal, the $N$ DFT outputs are just equally-spaced samples of $s_N[n]$'s DTFT. 
$s_N[n]$'s DTFT is related to $s[n]$'s DTFT; they may differ if $s[n]$ was longer than $N$ samples long originally and some truncation/windowing was involved in generating a finite sample length. 
$s[n]$'s DTFT is related to the original signal's Fourier transform, with the caveat that the DTFT is periodic, so all of the frequency content from the continuous-time signal that is unambiguously representable in discrete-time (the bandwidth of which is determined by the sample rate used) is contained within a single length-$2\pi$ period of $s[n]$'s DTFT.