What is an Algebra? And what is a unitary one? In my lecture notes, the following text appears:
"..We know that if $(X,d)$ is a metric space and $\mathbb K$ is $\mathbb R$ or $\mathbb C$, then $C(X,\mathbb K)$, that is the space of continuous function from $X$ to $\mathbb K$, equipped with the norm of uniform convergence, is a unitary algebra.."
I never came across the term algebra.

What does an "algebra" mean in this context? And what does "unitary algebra" mean? 

Don't bother yourself with a thorough explanation or something, I just need an $a$$b$$c$$d$.. list of the conditions that must be satisfied by a space of functions to be called an algebra, and what is the additional condition that makes it a unitary one. I know that the term "algebra" should not merely be a description for certain function spaces, it should describe a general vector space under certain conditions; but currently, I am just interested in what it means in this context and how the conditions are expressed in terms of functions for the purpose of direct application. 
Additionally:

When is a subspace called a "subalgebra"?

Again, I can relate to similar terms containing "sub" to guess that a subspace is called a subalgebra if it is an algebra by its own (with the induced operations from the initial space); but I need an $a$$b$$c$$d$.. list of what makes a subspace a subalgebra.
Thank you very much.
 A: An algebra $A$ over a field $\mathbb{F}$ is like $\mathbb{C}$ to the field $\mathbb{R}$.
It's a vector space $V$ over $\mathbb{F}$ and a mapping $\cdot : V \times V \rightarrow V$ linear in both arguments. An example is $n \times n$ matrices, with $\cdot$ being matrix multiplication. The complex numbers I've already said. The quaternions are an example.
An algebra is unitary if it has an $I$ such that $x \cdot I = I \cdot x = x  ,\forall x \in A$. The $I$ is called unity. All the examples above are unitary. But if the product operator is defined in such a way that it sends everything to $0$, then it's not unitary.
A space of continuous functions to a field is clearly an algebra because it's a vector space and the product operation is just multiplication of functions, which is clearly linear both left and right.
A subalgebra of an algebra is a vector subspace which is closed under the product operation.
A: If $A$ is a commutative ring, then an (associative, unital) $A$-algebra is a ring $B$ with a ring morphism $A\rightarrow B$ mapping $A$ to the center of $B$. (It is understood that $B$ is endowed with the restriction of scalars $A$-module structure.) The unitary requirement refers to the existence of a multiplicative identity (I take all rings to be unital), so this is implicit in my definition. The morphism $A\rightarrow B$ is called the structural morphism. An $A$-subalgebra of $B$ is a subring which contains the image of the structural morphism $A\rightarrow B$.
