Domains have an operational structure, that is, a collection of operations under which the domain is closed. For example, a group is closed under multiplication, taking the zeroth power of elements, and taking inverses of elements. The operational structure may be empty, examples of domains without additional structure are the underlying relations of general mappings (see Properties and Attributes of (General) Mappings).
The operations under which a domain is closed are a subset of the operations that the elements of a domain admit. It is possible that the elements admit more operations. For example, matrices can be multiplied and added. But addition plays no role in a group of matrices, and multiplication plays no role in a vector space of matrices. In particular, a matrix group is not closed under addition.
Note that the elements of a domain exist independently of this domain, usually they existed already before the domain was created. So it makes sense to say that a domain is generated by some elements with respect to certain operations.
Of course, different sets of operations yield different notions of generation. For example, the group generated by some matrices is different from the ring generated by these matrices, and these two will in general be different from the vector space generated by the same matrices, over a suitable field.
The other way round, the same set of elements may be obtained by generation w.r.t. different notions of generation. For example, one can get the group generated by two elements g and h also as the monoid generated by the elements g, g-1, h, h-1; if both g and h have finite order then of course the group generated by g and h coincides with the monoid generated by g and h.
Additionally to the operational structure, a domain can have properties. For example, the multiplication of a group is associative, and the multiplication in a field is commutative.
Note that associativity and commutativity depend on the set of elements for which one considers the multiplication, i.e., it depends on the domain. For example, the multiplication in a full matrix ring over a field is not commutative, whereas its restriction to the set of diagonal matrices is commutative.
One important difference between the operational structure and the properties of a domain is that the operational structure is fixed when the domain is constructed, whereas properties can be discovered later. For example, take a domain whose operational structure is given by closure under multiplication. If it is discovered that the inverses of all its elements also do (by chance) lie in this domain, being closed under taking inverses is not added to the operational structure. But a domain with operational structure of multiplication, taking the identity, and taking inverses will be treated as a group as soon as the multiplication is found out to be associative for this domain.
The operational structures available in GAP form a hierarchy, which is explicitly formulated in terms of domain categories, see Domain Categories.
GAP 4 manual