If we take "algebraic structure" to be a synonym for "algebra" (in the sense of universal algebra), then an algebraic structure is a set $X$, together with a family of operations on $X$.
Recall that given a set $X$, an "operation" on $X$ is a function $X^{\alpha}\to X$, where $\alpha$ is an ordinal. Such a function is called an $\alpha$-ary operation; when $\alpha$ is a natural number, the operation is said to be "finitary" (takes only finitely many arguments).
Sometimes, algebraic structures are further enriched with (i) "partial operations" (functions defined on a subset $A\subseteq X^{\alpha}$ rather than all of $X^{\alpha}$), or (ii) $\beta$-ary relations (subsetes of $A^{\beta}$). We can also impose identities (requires that the operations/relations satisfy certain properties such as commutativity, etc).
In this sense, vector spaces, groups, rings, fields, etc. are all (enriched) "algebras"; metric spaces are not.
"Space" is a bit fuzzier; I would not put "vector spaces" in the class, restricting it rather to things like topological spaces, manifolds, metric spaces, normed spaces, etc.
Now, one should realize that you this does not have to be a dichotomy: you can have structures that include both kinds of data: a topological group is both an algebraic structure (a group) and a space (topological space), in a way that makes both structures interact with one another "nicely". Normed vector spaces are both algebraic structures (vector spaces), and "spaces" (normed spaces, hence metric, hence topological), where, again, we ask that the two structures interact nicely.
In fact, there is a lot of interesting stuff that can be obtained by having the two kinds of structures and "playing them off against one another." For example, Stone Duality and Priestley Duality exploit this kind of "structured topological space" (a topological space that also has operations, partial operations, and relations that interact well with the topology).
Let $G$ and $H$ simply be sets filled with numbers. We can easily define mappings between $G$ and $H$. Let's say that $f : G \to H$ is given by $f(g) = g * g = g^2$. We could also say that $f(g) = g^3$ or $f(g) = \frac{57}{g}$. These are all "mappings". A "mapping" is simply a rule that one uses to take an element of $G$ and "mess around" with it to get something out. Imagine a machine, that has an input and an output. Nothing is particularly special about a mapping, it is simply a rule as stated above.
A $\textit{Morphism}$ is a much more interesting kind of map. Let's endow $G$ and $H$ with group structure and call them $(G, *)$ and $(H, \diamond)$. Now we ask the question: can we look at some mapping, $\phi$ say, that $\underline{\textbf{preserves}}$ the group structure? That is to say, if $\phi : G \to H$, then to be a $\textit{homomorphism}$ we require that for $g, h \in G$
$$\phi(g*h) = \phi(g) \diamond \phi(h).$$
Now, several properties of the structure follow from this. If $e_G$ is the identity in $G$ then $\phi(e_G) = e_H$ in $H$. In English, we can state that via a group homomorphism, identities map to identities. Other properties of the structure are also preserved under the morphism.
There are several morphisms that have different requirements depending on your context. That is to say, Ring Homomorphisms differ from Group Homomorphisms, Linear Transformations are actually the morphisms of the category of Vector Spaces, the morphisms of topological spaces are continuous maps etc. etc.
LET IT BE SAID, however, that the terms "mapping" and "function" are often used interchangeably by different people and the description given above is that of my own usage of the term.
Best Answer
A helix is a curve in 3-d space with an axis, where the tangent line makes a constant angle with the axis. See here for some info on the helix. A circle would be the only helix which could lie in a plane, since the tangent line to a plane curve is always at 90 degrees from an axis normal to that plane. So a 2-d helix is not much of a helix, which more typically resemble corkscrews.
[Note: I take that back about plane curves --- the tangent line to any curve in the $xy$ plane (at any point on it) is at a 90 degree angle from the $z$ axis, so that any curve in the $xy$ plane would qualify as a helix under the constant angle definition. That makes such curves not good candidates for the term ""helix", i.e. the notion is trivially true for all plane curves.]
Spirals on the other hand do not need to have that property, and are typically curves in a 2-d plane. See the link here for some information about various types of spiral. Some of them are quite interesting.