The Rayleigh-Plesset equation is the fundamental model of bubble dynamics and is widely used in the study of cavitation mechanisms. Since cavitation bubble often occurs at the micro-scale, the effect of surface tension will have an important influence on the bubble motion. Based on the Sundman transformation and Weierstrass elliptic function theory, this paper studies the Rayleigh-Plesset equation considering surface tension, and establishes the parametric theoretical solutions of vapor bubble and gas bubble respectively. The results show that the vapor-bubble and gas-bubble dynamic equations can be solved theoretically. Further, based on the theoretical solutions, the effects of the surface tension on the motion of vapor bubbles and gas bubbles are studied in detail. For the two types of bubbles, the influence of the surface tension on the bubble motion will increase gradually as the scale of the bubble gets smaller. Especially when the bubble scale is reduced to 10 µm, the effect of the surface tension becomes too significant to be negligible.