I am confused about the difference between the tenth part or tithe(depending on the translation) and charity. Since there is no temple, does the tenth part offering go to the local synagogue? And do the levels of Tzedakah suggested by Rambam still apply today?

The tithing the Torah is speaking about is giving ten percent of the crops to the Levites and ten percent more to give to the poor or to take and eat in Yerushalaim – depending on what year it is. That applied only in Israel and only in time when a Levite could prove his lineage, that is not the case today so the Ma'aser – tithe of that implication does not apply nowadays. However there is a Mitzvah of Tzedakah that every Jew has to give charity to the poor – with the intention to bring him back on his feet financially, as the Rambam and Shulchan Aruch teach. An average amount for the average person should be ten percent of the annual income. If one can't afford that because his financial difficulties – he can give less. One that can afford more – may give more.