The inheritance of acquired characteristics is commonly referred to as Lamarckism. Charles Darwin himself considered it a possible element in providing the raw material for natural selection to work on, since genetics hadn't been developed yet.
Lysenko convinced Stalin that using Lamarckist principles could lead to faster adaptation of plants to harsh climates, which was obviously a tempting thought in Russia. It probably did less harm than Khrushchev's Virgin Lands project, draining the minimal Central Asian watercourses to grow cotton in an inhospitable desert. But Stalin was a "bad" Russian and Khrushchev was a "good" Russian, so Lysenko gets the blame.