Kansai-ben is one of the most well-known dialects of Japan, being frequently used on television to give certain characters a more country bumpkin feel or elderly appeal. The reason for its popularity primarily has to do with the fact that Kansai homed the capital of Japan in Kyoto until 1869, when power was shifted over to Tokyo. Today, the Kansai region still remains one of the most prosperous and most traveled areas of Japan.
Dialectally, Kansai falls into the Western Japanese grouping. So we see lots of traits shared with other Western areas, such as negation in -(a)n rather than -(a)nai: わからん wakaran, not わからない wakaranai "not know; not understand". Another popular feature, though now strongly in regression, is the adverbial form in -u (+coalescence) rather than -ku, so we would get はよう hayou (from はやう hayau), instead of はやく hayaku. Though this latter difference may seem strange, it remains preserved in the standard language in certain set expressions and forms of honorific speech, such as おはよう ohayou, whose true equivalent would technically be おはやく ohayaku, or おめでとう omedetou, instead of おめでたく omedetaku. It's possible that these remained in use because the dialect of the former capital was still regarded as being proper or even prestigious, and so this was reflected in its limited borrowing into the Tokyo area.
Some features, however, are fairly unique to the region. We see, for example, the copula "to be" being reduced to じゃ ja and や ya throughout the region (including derived froms like じゃろう jarou and やろう yarou), with the polite equivalent being だす dasu or どす dosu depending on the area. Adjectives commonly drop the final -i and extend the vowel of the last sound, as in さむー samuu rather than さむい samui "it's cold". But it's not uncommon to hear the final -i coalesce (merge) with the preceding vowel either, so さむい samui could very well become さみい samii as well. Typically, a sequence of /u+i/ will become /ii/, while /a+i/ or /o+i/ will become /ee/.
There are also a lot of morphological changes to the verbal and adjectival paradigms that I alone cannot explain. But some noticeable differences might be formal negation, where the familiar -masen form of standard Japanese becomes -(a)hen (or -(a)hin) in Kansai. For example, 行きません ikimasen "not go" becomes 行かへん ikahen; しません shimasen "not do" irregularly becomes せえへん seehen or しいひん shiihin; 見ません mimasen "not look" becomes めえへん meehen or 見いひん miihin; and so on.