  By Mazur's Torsion Theorem, there are fourteen possibilities for the
non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, 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 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~$1$.
