A plea of NO CONTEST simply means that you do not wish to contest the City's charge against you, but wish to talk to the Judge about mitigating circumstances. Judgment will be entered by the Judge and some penalty will be set. A plea of no contest cannot be used against you in a civil suit for damages.
Plead the fifth is a reference to the Bill of Rights. "No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation." Basically, you have the right to not make yourself guilty by saying something stupid.