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The book I'm reading gives the definition of a simply connected space as:

X is simply connected iff X is arcwise connected and $\pi(X,x) = \{1\}$ for some (and hence any) $x$.

What does a fundamental group $=\{1\}$ mean? I understand that the fundamental group is a set of equivalence classes of loops about a point $x$, but I have never seen this notation.

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    $\{1\}$ is the trivial group, the one containing only the identity – 2017-02-21
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    So since the fundamental group is a partition of all loops at x and all the loops are in the same equivalence class, then all loops at x are equivalent? – 2017-02-21
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    yes, see Bill's answer for more details – 2017-02-21

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This means that there is only 1 equivalence class of loops. That class would include the trivial (constant) loop: $f(t)=x_0$ for all $0 \leq t \leq 1$ (where $x_0$ is the base point for $\pi(X,x_0)$).

In other words, every loop $g:[0,1] \to X$ (where $g$ is continuous and $g(0)=g(1)=x_0$) can be deformed continuously to the constant loop $f$. This means that there is a continuous map $H:[0,1] \times [0,1] \to X$ which starts as $g$: $H(0,t)=g(t)$ for $0 \leq t \leq 1$, ends at $f$: $H(1,t)=f(t)=x_0$ for $0 \leq t \leq 1$, and is a bunch of loops in between: $H(s,0)=H(s,1)=x_0$ for $0 \leq s \leq 1$.