The "middoth" were Rabbi Hillel's rules for interpretation. There are seven of them. What are they? Please list them and include one or two clear examples for each.
Hillel's seven were later expanded by Rabbi Yishmael ben Elisha into thirteen, dropping one. I'll be drawing this list from the list on Wikipedia. I'll be drawing some examples from this answer on Mi Yodeya about R. Yishmael's list that in turn quotes from the Artscroll prayer book.1
Kal v'chomer: if a lenient case has a stringency, then surely a stringent case has it. For example, the set of forbidden labors on Shabbat is a superset of those forbidden on yom tov (festivals plus Rosh Hashana and Yom Kippur). So if a labor is forbidden on yom tov, which has more-lenient rules, then surely it is forbidden on Shabbat.
G'zeirah shavah: two ideas are related through commonality in one of two ways, either (a) the same word is used in two different places (so we understand similar meaning) or (b) two adjacent verses are related. An example of (a): the daily offering is brought even on Shabbat and from this we learn that the Pesach offering is brought even on Shabbat because the same word is used to describe the time (b'moado). An example of (b) (quoting from the Mi Yodeya answer): the law that one could get married through a contract is derived from the verse "She left from his house, and became married to another". Through this we learn that "leaving" is compared to "marrying", and just like a divorce is done through a contract, so too could marrying.
Binyan av mi-katuv echad: a decision based on one passage applies to similar passages. Example: marrying one's mother's half-sister is forbidden, and we derive that marrying one's father's half-sister is also prohibited.
Binyan av mi-shene ketuvim: a decision in two laws having a common characteristic is applied to other laws with that characteristic. I think (but am not sure) that this rule means that if we know that, say, weaving and writing are both forbidden on Shabbat because they are elements of making the mishkan (and building the mishkan on Shabbat was forbidden), then we can also say that dyeing, cooking, and several others that are also part of mishkan-building are forbidden. (This example is my own, not Artscroll's.)
Klal u'prat and prat u'klal: Complicated rules about relating the particular to the general and the general to the particular. One example (from Mi Yodeya), one who kills someone is executed (no differentiation between careless or premeditated murder). When the Torah later on refers to the punishment of exile for a careless murderer, it cannot be adding to execution, but to replace it with exile. For this one I'm going to point to the sources I've linked for more details.
Ka-yotzei bo mi-makom acher: explaining a biblical passage according to another of similar content. R. Yishmael omitted this one and I don't have an example.
Davar halamed me-inyano: proving something from context. For example, one of the commandments at Sinai is "do not steal". This comes after "do not commit adultery" and "do not murder", both of which are capital crimes, so "do not steal" is understood to refer to kidnapping because that is the only capital form of stealing.
1 A prayer book (siddur) may seem an unusual source for rules of rabbinic interpretation. Jews are commanded to study torah daily, so some torah study is built into the morning service -- including R. Yishmael's 13 rules for interpretation. Artscroll, a major publisher whose siddur is used widely, augments the list with examples, as noted at the link.
See this article for some additional information (starting near the bottom of the first page).
Please note: This answer was written for a neutral, academic audience and is not intended to be interpreted in the context of a religious belief or doctrine.