A raceme is a type of inflorescence that is unbranched, indeterminate, and bears pedicellate flowers (having short floral stalks called pedicels) along the axis. In botany, axis means a shoot, in this case one bearing the flowers. In a raceme, the oldest flowers are borne near the base.
A spike is a type of raceme in which the individual flowers are sessile (that is, lack pedicels). The term spikelet can refer to a small spike, although is also used to refer to the ultimate flower clusters in the grasses (Family Poaceae) and sedges (Family Cyperaceae), in which cases the stalk supporting the cluster becomes the pedicel. A spadix is a form of spike in which the florets are densely crowded along the axis.