More in general with Van Kampen theorem one have to deal with notation of the following kind
$\langle g_1,\dots,g_s \mid m_1(g_1,\dots,g_s) = n_1(g_1,\dots,g_s);\dots; m_k(g_1,\dots,g_s) = n_k(g_1,\dots,g_s) \rangle$ where $m_i(g_1,\dots,g_s)$ and $n_j(g_1,\dots,g_s)$ are strings containing as symbols $g_i$ or $g_i^{-1}$.
This notation is called a finite presentation of a group: it simply denote the biggest group having $s$ generators, named $g_1,\dots,g_s$ for which the relations
$m_i(g_1,\dots,g_s) = n_i(g_1,\dots,g_s)$ hold. More formally the notation above denote the quotient group of the free group with $s$-generators by the smaller normal subgroup containing the elements of the form
$m_i(g_1,\dots, g_s) \left(n_i(g_1,\dots,g_s)\right)^{-1}\ \text{.}$
In this group the equalities $m_i=n_i$ hold and every other group for which $m_i=n_i$ is a quotient of this group, thus the attribute bigger.
Some additional notes: in your examples $\langle a \mid a = 1 \rangle$ means the group with one generator with is equal to the identity, the group with this property is the trivial group $\{1\}$ that contain just the identity (so this proves that you space is indeed simply connected. In the case of the torus $\langle a,b \mid aba^{-1}b^{-1} = 1\rangle$ is also the group $\langle a,b \mid ab=ba\rangle$, this group is the biggest group having two generators which commute, i.e. it is the free abelian group with two generators.