Let $X$ be a total space. If we let symmetric difference take the place of addition: $A \Delta B = (A \cup B) - (A \cap B)$, and let intersection be multiplication, then a sigma algebra of subsets of $X$ becomes a boolean ring with the empty set being $0$ and the total space being $1$. In general a family of sets closed under intersection and symmetric difference, and having the emptyset is a called a ring of sets and is actually a ring (or for some, a rng since it need not have unity), in addition it is a Boolean ring of characteristic 2 as for any $A$ we have $A+ A = \emptyset$ and $A^2 = A$.
An algebra of sets is a ring containing the total space, or in other words, it is a ring with unity, note that this is equivalent to closing the family under complements.
The notion of $\sigma$-rings/algebras corresponds to a more measure theoretic desire for countable operation, although they still behave well since all operations are Abelian.
Note that a $\sigma$-algebra with addition and multiplication given by symmetric difference and intersection is actually a (unital) algebra over the field $\mathbb{Z}/2\mathbb{Z}$, which can be imbedded in the algebra as the set $\{\emptyset,X\}$
Edit: Looking back at your question, you also expressed some confusion as to what an algebra is. For most things you'll see an algebra is a set $A$ that is at once equipped with the structures of a ring and vector space (or module) over a field ( or ring) $K$. However we want ring addition and vector addition on $A$ to coincide, and we want ring multiplication to be bilinear, which really just boils down to saying that for $v,w \in A$ and $\lambda \in K$, we want $$(\lambda v)\cdot w = v \cdot(\lambda w) = \lambda(v \cdot w)$$
If you're comfortable with rings, an elegant way of phrasing this in that context is simply that a unital algebra is a ring $A$ together with a ring of scalars $R$ and homomorphism of rings $f: R \to A$. Scalar multiplication is just $\lambda v = f(\lambda) \cdot v$
Best Answer
An algebra over a field is like a vector space with some sort of multiplication between vectors, like 3-dimensional real space with the cross product.
A field is like a set with some notion of addition, subtraction, multiplication and division, like the field of real numbers.
Every field is an algebra because every field is a (one dimensional) vector space, but not every algebra is a field. The previous example of real 3-dimensional space with the cross product is such an algebra.