We perform resistive magnetohydrodynamic simulations to study the internal structure of current sheets that form during solar eruptions. The simulations start with a vertical current sheet in mechanical and thermal equilibrium that separates two regions of the magnetic field with opposite polarity which are line-tied at the lower boundary representing the photosphere. Reconnection commences gradually due to an initially imposed perturbation, but becomes faster when plasmoids form and produce small-scale structures inside the current sheet. These structures include magnetic islands or plasma blobs flowing in both directions along the sheet, and X-points between pairs of adjacent islands. Among these X-points, a principal one exists at which the reconnection rate reaches maximum. A fluid stagnation point (S-point) in the sheet appeared where the reconnection outflow bifurcates. The S-point and the principal X-point (PX-point) are not co-located in space though they are very close to one another. Their relative positions alternate as reconnection progresses and determine the direction of motion of individual magnetic islands. Newly formed islands move upward
