This is a long comment on André's answer. Here is a way to organize the spaces that appear in it using the categories of $\mathbb{Z}_2$-graded modules over Clifford algebras.
Let $\text{Cliff}(p, q)$ denote the Clifford algebra obtained by adjoining $p$ anticommuting square roots of $1$ and $q$ anticommuting square roots of $-1$ to $\mathbb{R}$, and let $\text{Mod}(n)$ denote the category of $\mathbb{Z}_2$-graded modules over $\text{Cliff}(p, q)$ for any $p, q$ such that $p - q \equiv n \bmod 8$ (this is part of the Clifford-algebraic statement of Bott periodicity). Then:
- $\text{Mod}(0) \cong \text{Vect}_{\mathbb{R}} \times \text{Vect}_{\mathbb{R}}$.
- $\text{Mod}(\pm 1) \cong \text{Vect}_{\mathbb{R}}$.
- $\text{Mod}(\pm 2) \cong \text{Vect}_{\mathbb{C}}$.
- $\text{Mod}(\pm 3) \cong \text{Vect}_{\mathbb{H}}$.
- $\text{Mod}(\pm 4) \cong \text{Vect}_{\mathbb{H}} \times \text{Vect}_{\mathbb{H}}$.
Each of these module categories is in particular symmetric monoidal under direct sum and so presents an infinite loop space, which I'll label $K(n)$. These infinite loop spaces are the following:
- $K(0) \cong KO \times KO$.
- $K(\pm 1) \cong KO$.
- $K(\pm 2) \cong KU$.
- $K(\pm 3) \cong KSp$.
- $K(\pm 4) \cong KSp \times KSp$.
These categories and these infinite loop spaces can be organized into a clock in the obvious way. There are natural pairs of adjoint functors given by induction and restriction between any consecutive categories in this clock, and these induce maps on the corresponding infinite loop spaces. The following statement implies Bott periodicity in the strong form that André states it (which is the form in which Bott originally proved it).
Theorem: The homotopy fiber of the natural map $K(n) \to K(n-1)$ is also the homotopy cofiber of the natural map $K(n+1) \to K(n)$.
Corollary: The homotopy fiber of the natural map $K(n) \to K(n-1)$ and the homotopy cofiber of the natural map $K(n+1) \to K(n)$ are both the $n$-fold loop space $\Omega^n KO$, and $\Omega^8 KO \cong KO$.
On the other hand, you should be able to convince yourself at least heuristically that the homotopy fiber of the natural map $KO \times KO \to KO$ is $KO$, that the homotopy fiber of the natural map $KU \to KO$ is $O/U$, and so forth. In other words, you can rewrite that sequence of spaces heuristically as
$$\frac{KO \times KO}{KO}, \frac{KO}{KO \times KO}, \frac{KU}{KO}, \frac{KSp}{KU}, \frac{KSp \times KSp}{KSp}, \frac{KSp}{KSp \times KSp}, \frac{KU}{KSp}, \frac{KO}{KU}.$$
where $\frac{X}{Y}$ is heuristic notation for the homotopy fiber of a map $X \to Y$, and in particular $\frac{1}{BG}$ is $G$, so that $\frac{KU}{KO}$ becomes $O/U$ and so forth.