Simplicial Complex vs Delta Complex vs CW Complex

Simplicial complexes, $\Delta$-complexes, and CW-complexes are all constructed by gluing together simplices. However, for each one, there are different rules for what kinds of "gluings" you are allowed to use.

For CW-complexes, you are allowed to use almost any gluing. Specifically, a CW-complex is constructed by induction, where at each step, you adjoin a new simplex by gluing its boundary to the complex you have already by any map. More explicitly, if $Y$ is the CW-complex you have built so far and $f:\partial \Delta^n\to Y$ is any continuous map, you can build a CW-complex $X=Y\sqcup\Delta^n/{\sim}$, where $\sim$ is the equivalence relation that identifies any $x\in\partial\Delta^n$ with $f(x)\in Y$. The only restriction to this gluing process is that you have to add simplices in increasing order of dimension. That is, you have to start with all the $0$-simplices, then glue in all the $1$-simplices, then glue in all the $2$-simplices, and so on. You're not allowed to glue in a new $1$-simplex once you've already added a $2$-simplex. (If you drop this ordering condition, you get a more general notion, which is sometimes called simply a "cell complex".)

For $\Delta$-complexes, you do the same thing, except that the maps $f$ you can use when adding a new cell are highly restricted. Specifically, for each $(n-1)$-simplex $A$ which is a face of $\partial\Delta^n$, the restriction of $f$ to $A$ must be equal to the inclusion of one of the $(n-1)$-simplices you already have. That is, $f$ maps the $n$ vertices of $A$ (with their canonical ordering) to the $n$ vertices of some $(n-1)$-simplex you've already added to your complex (with the same ordering on the vertices), and $f$ extends to all of $A$ by just interpolating linearly. Intuitively, this means that your complex is a union of simplices which are glued together by just gluing their faces together in the "obvious" linear way (for instance, as one encounters in a triangulation of a surface), rather than by arbitrary complicated continuous maps. Note, however, that some faces of a single simplex might get glued to each other: the restriction on what $f$ can be only applies to each face of $\partial\Delta^n$ separately. So, for instance, you can start with a single vertex, and then add an edge both of whose boundary vertices are the one vertex you started with (this gives you a circle). You could then add a triangle such that each of its three sides are equal to the one edge you have (this gives a space which cannot be embedded in $\mathbb{R}^3$ and is rather hard to visualize!).

Finally, simplicial complexes are $\Delta$-complexes which satisfy even more restrictions. First, $f$ is required to map different faces of $\partial\Delta^n$ to different $(n-1)$-simplices, so the situation discussed at the end of the previous paragraph cannot happen. In addition, you are not allowed to add two different $n$-simplices with the same set of vertices, so that a simplex in a simplicial complex is uniquely determined by its set of vertices (which, by the first requirement, you can show are all distinct).