For the record, the letters affected in Modern Hebrew by the dagesh are Bet, Cahf and Peh. There are also two types of dageshes - a “hard” dagesh and a “soft” dagesh - but you really don’t want me to go there.
Now, for those of you who are having problems understanding this whole “Ashkenazic” and “Sephardic” Hebrew thing, it’s more or less like this. After the exiles from Israel, most Jews stopped speaking Hebrew (or Aramaic) on a day to day basis, instead adopting the languages of whatever country there settled down in. With time, several unique Jewish dialects developed, but Hebrew itself was relegated to a largely ceremonial role, much like Latin in the middle ages (although the majority of Jewish males spoke some degree of Hebrew). Because it was no longer anyone’s mother tongue, people generally started speaking Hebrew in whatever accent they grew up speaking.
Eventually, two major Hebrew dialects appeared - Ashkenazic, which was spoken in Europe, and was based mainly upon German, and Sephardic, which was based on Arabic. Both dialects were grammatically identical (because they both relied on the same written sources), but were nearly mutually incomprehensable. Not only were consonants and vowels pronounced differently, but different syllables were emphasized: while Ashkenazic Hebrew tends to stress the beginning of words, Sephardic always emphasises the end. Thus, a European Jew would pronounce “Happy New Year” as SHA-na TO-va, an Oriental Jew would say sha-NA to-VA.
When Eliezer Ben Yehuda revived the Hebrew language as a secular tongue at the turn of the century, he decided to adopt Sephardic pronounciation - this, despite his own Ashkenazic background. As a linguist, he decided that Arabic, being a fellow Semitic language, was probably much closer to the way that Hebrew was originally pronounced than German, which bears it no relation. So the new Zionist olim started speaking their Tahfs with a hard “T”, and pronouncing their Patachs as “ah” instead of “oh” or “oy”. The problem was, these new immigrants were nearly all from Russia, Poland and Germany. There was no way they could pronounce Sephardic Hebrew’s gutteral Eins and Chets, which were very similar to those of the neighbouring Arabs. They also let some of their native accents slip into their speach (along with a great deal of Yiddish slang). The result is sort of a “Sphardic Light”, a completely new dialect that may be closer to Sephardic than Ashkenazic, but can stand on its own - and also has a much larger vocabulary than either.