There is a minimal way to imbed a nonunital $C^*$-algebra $A$ into a unital $C^*$-algebra. As a $*$-algebra this is $A\oplus \mathbb C$ with componentwise addition, with multiplication $(a,s)(b,t)=(ab+ta+sb,st)$, and with involution $(a,s)^*=(a^*,\overline s)$. But to give this algebra a $C^*$ norm, one method is to identify it with $\{L_a:a\in A\}+\mathbb C\mathrm{id}_A\subset B(A)$, where $L_a:A\to A$ is defined by $L_ab=ab$. One can then check that the operator norm of this algebra as a subspace of $B(A)$ is a $C^*$ norm.
There is also a maximal way to imbed a nonunital $C^*$-algebra $A$ into a unital $C^*$-algebra as an ideal in an "essential" way. The essentialness is captured by stipulating that every nonzero ideal in the unitization intersects $A$ nontrivially. This is equivalent to the condition that $bA=\{0\}$ implies $b=0$. As mland mentioned, this maximal unitization is the multiplier algebra of $A$, $M(A)$. The double centralizer approach is one particular concrete description, but $M(A)$ has other decriptions and is characterized by a universal property: For every imbedding of $A$ as an essential ideal in a $C^*$-algebra $B$, there is a unique $*$-homomorphism from $B$ to $M(A)$ that is the identity on $A$.
t.b. has already mentioned that in the commutative case this runs parallel to one-point versus Stone–Čech compactification.
Here is another example. The algebra $K(H)$ of compact operators on an infinite dimensional Hilbert space $H$ has minimal unitization (isomorphic to) $K(H)+\mathbb CI_H$, and multiplier algebra (isomorphic to) $B(H)$.
One reason we may want to go all the way to $M(A)$ is to better understand automorphisms of $A$. Conjugation by a unitary element of $M(A)$ is an automorphism of $A$. In the case of $K(H)\subset B(H)\cong M(K(H))$, every automorphism is of this form, and you couldn't get most of these automorphisms by only conjugating by unitaries in the minimal unitization $K(H)+\mathbb C I_H$.
The approach mentioned by mland of identifying $M(A)$ with the algebra of adjointable operators on $A$ can be found in Lance's Hilbert C*-modules or in Raeburn and Williams's Morita equivalence and continuous trace C*-algebras with a lot more useful introductory information in each. I agree with mland that for the basics of K-theory you do not need to get into multiplier algebras, but you can learn more about their importance in K-theory from Blackadar's K-theory for operator algebras. Chapter VI is described as a collection of "all the results needed for Ext-theory and Kasparov theory," and it starts with a review of multiplier algebras and examples.