  By Mazur's Torsion Theorem, there are fourteen possibilities for the
non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$,
such that $E$ may have additive reduction at a prime $p$, we consider a
parameterized family $E_T$ of elliptic curves with the property that they
parameterize all elliptic curves $E/\mathbb{Q}$ which contain $T$ in their
torsion subgroup. Using these parameterized families, we explicitly classify
the Kodaira-N\'{e}ron type, the conductor exponent, and the local Tamagawa
number at each prime $p$ where $E/\mathbb{Q}$ has additive reduction. As a
consequence, we find all rational elliptic curves with a $2$-torsion or a
$3$-torsion point that have global Tamagawa number $1$.
