Smash product

Naive definition
Given two based topological spaces $$(X,x_0)$$ and $$(Y,y_0)$$, their smash product denoted $$(X \wedge Y,*)$$ is defined as the quotient of $$X \times Y$$ by the following equivalence relation:

$$(x,y_0) \simeq (x_0,y)$$

In other words, we collapse both the copy of $$X$$ and the copy of $$Y$$, through $$(x_0,y_0)$$, to a single point (the union of the $$X$$-copy and $$Y$$-copy is isomorphic to the wedge sum of the spaces, hence the smash product can be viewed as the quotient of the product by collapse of the wedge sum to a point).

Category-theoretic definition
The smash product is the natural notion of product in the category of based topological spaces. Category-theoretically, this means that given maps of based topological spaces to $$X$$ and $$Y$$ from $$Z$$, there exists a unique map to $$X \wedge Y$$ from $$Z$$ such that the diagram commutes.

Reduced suspension
The reduced suspension of a topological space is its smash product with the based circle (since the circle is homogeneous, it does not matter what basepoint we choose for it. The reduced suspension is often just called the suspension.

Spheres
The $$n$$-sphere (with basepoint) can be viewed as the smash product of the (based) circle with itself $$n$$ times.