What goes into a vaccine placebo, typically and in the specific case of Pfizer's SARS-CoV-2 vaccine trials?
Simple answer: This vaccine trial compared their trial vaccine against saline.
See the original publication by Biontech and Pfizer in the New England Journal of Medicine linked below. The relevant information can be found in the methods section:

TRIAL PROCEDURES
With the use of an interactive Web-based system, participants in the trial were randomly assigned in a 1:1 ratio to receive 30 μg of BNT162b2 (0.3 ml volume per dose) or saline placebo. Participants received two injections, 21 days apart, of either BNT162b2 or placebo, delivered in the deltoid muscle. Site staff who were responsible for safety evaluation and were unaware of group assignments observed participants for 30 minutes after vaccination for any acute reactions.

As far as I followed up the case of the allergic reactions, two of these have been observed in England. Both people had an epipen injector for such cases by hand, making it possible that they have underlying problems with allergies (as this is not a medical device people usually have at hand), no such cases have been reported from the trial. Allergic reactions are always a small risk with vaccinations which is the reason why this is done in a controlled medical environment.
As for the vaccine reactions in the placebo group, this can be caused by the nocebo effect, where expecting negative effects leads to a more negative reaction. As @BryanKrause mentioned, all possible effects reported in such trials are recorded as possible side effects. This leads to a lot of things being recorded which happened at the same time as the vaccination but have no connection to it.
Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine