We present results from the first combined study of variable stars and star formation history (SFH) of the Milky Way (MW) "ultra-faint" dwarf (UFD) galaxy Leo T, based on F606W and F814W multi-epoch archive observations obtained with the Wide Field Planetary Camera 2 on board the Hubble Space Telescope. We have detected 14 variable stars in the galaxy. They include one fundamental-mode RR Lyrae star and 10 Anomalous Cepheids with periods shorter than 1 day, thus suggesting the occurrence of multiple star formation episodes in this UFD, of which one about 10 Gyr ago produced the RR Lyrae star. A new estimate of the distance to Leo T of 409 $^{+29}_{-27}$ kpc (distance modulus of 23.06 $±$ 0.15 mag) was derived from the galaxy's RR Lyrae star. Our V, V-I color-magnitude diagram of Leo T reaches V~29 mag and shows features typical of a galaxy in transition between dwarf irregular and dwarf spheroidal types. A quantitative analysis of the star formation history, based on the comparison of the observed V,V-I CMD with the expected distribution of stars for different evolutionary scenarios, confirms that Leo T has a complex star formation history dominated by two enhanced periods about 1.5 and 9 Gyr ago, respectively. The distribution of stars and gas shows that the galaxy has a fairly asymmetric structure. ; Comment: 31 pages, 14 figures, accepted for publication in ApJ