  Detailed measurements of the global and local electromagnetic properties of
Ba(Fe$_{0.93}$Co$_{0.07}$)$_{2}$As$_{2}$ single crystals are reported. Analysis
of the irreversible magnetic response provides strong evidence for similar
vortex physics in this Fe-based pnictide superconductor to the high-$T_{c}$
cuprates, such as Y-Ba-Cu-O or Nd-Ce-Cu-O. In particular, we have found a
nonmonotonic "fishtail" magnetization in $M(H,T=const)$ loops and its signature
is also present in $M(H=const,T)$ scans. The supercurrent density is evaluated
by using several techniques, including direct transport measurements. At 5 K we
estimate its value to be a moderate $j \approx 2.6 \pm 0.2 \times 10^5$
A/cm$^2$. Analysis of the magnetic relaxation is consistent with the collective
pinning and creep models (weak pinning and fast creep) and suggests a crossover
from the collective to the plastic creep regime in fields exceeding the value
corresponding to the maximum in fishtail magnetization.
