Meaning of channel capacity in the AWGN channel
There are two different kinds of channel models that are being confused here.

In the binary symmetric channel, the inputs and the outputs
are constrained to be $0$ or $1$ and the key parameter is the
transition probability: the probability that an input $0$ is changed
to an output $1$ or vice versa.  The channel capacity $C$
is a number between $0$ and $1$ and the meaning is that by using
a suitable code of rate $R &lt; C$ on the channel, we can transmit
at a rate of $R$ information bits per channel use with arbitrarily
high reliability (that is, with arbitrarily small bit error
probability.  A (block) code for such a channel would be a set
of $M$ binary $n$-tuples that will convey $\log_2 M$ bits in $n$ channel
uses at a rate $R = \frac{\log_2 M}{n}$. Usually $M$ is chosen as
a power of $2$, say $2^k$ for some $k &lt; n$.  Note also that the
binary symmetric channel capacity cannot exceed 1 bit per channel use.

The discrete-time Gaussian channel is a different beast. The inputs 
can be arbitrary real numbers (it is not necessary that they 
be restricted to $0$ or $1$) and if the $i$-th input is $x_i$,
then the $i$-th output is $x_i+n_i$ where the $n_i$ are independent
zero-mean Gaussian random variables with variance $\sigma^2$.
$x_i^2$ is called the power in the $i$-th input and
$\sigma^2$ is called the noise power.  A (block) code for such
a channel is a collection of $M$ $n$-vectors $\mathbf x^{(i)}$
(called a codebook) that can be used
to transmit $\log_2 M$ bits in $n$ channel uses.  Various realistic 
constraints are placed on the code.  One 
such constraint is that the peak power is bounded by $P$:
the magnitudes of all the $M\times n$ entries $x_j^{(i)}$
in the codebook are no larger than $\sqrt{P}$.
Another is that 
that the average power is bounded by $P$: 
$$\frac{1}{Mn}\sum_{i} ||\mathbf x^{(i)}||^2 
= \frac{1}{Mn} \sum_i \sum_j \left|x_j^{(i)}\right|^2\leq P.$$
The capacity of the channel depends on the signal to noise ratio
$P/\sigma^2$ and it can exceed one bit per channel use.  Think of it
this way: we can use multilevel pulse amplitude modulation to transmit
more than one bit per channel symbol on this channel whereas with
the binary symmetric channel we are restricted to binary modulation
and so cannot exceed one bit per channel use.

For lots of details on this kind of stuff at a level lower than
a typical graduate text, see the following links (lecture notes
of mine).

Lecture 6,
Lecture 7,
Lecture 8,
Lecture 9,
Lecture 10,
Lecture 11,
Lecture 12