How do you make a call on an Amateur Repeater?
First, LISTEN AND LISTEN SOME MORE...... to make sure that the repeater is not already in use. When you are satisfied that the repeater is not in use, set your transmitter power to the minimum and increase only as needed to make contact with the repeater, begin with the callsign of the station you are trying to contact followed by your callsign. e.g. " N4??? this is N3???". (The N3??? is your callsign). If you don't establish contact with the station you are looking for, wait a minute or two and repeat your call.
If you are just announcing your presence on the repeater it is helpful to others that may be listening if you identify the repeater you are using AND your callsign. e.g. " This is N3??? listening on the 84 machine or you could also say This is N3??? listening on 146.84 Dallas or the location of the repeater if known.  This allows people that are listening on radios that scan several repeaters to identify which repeater you are using.
If the repeater you are using is a busy repeater you may consider moving to a simplex frequency (transmit and receive on the same frequency..... see more below), once you have made contact with the station you were calling. Repeaters are designed to enhance communications between stations that normally wouldn't be able to communicate because of terrain or power limitations.
If you can maintain your conversation without using the repeater, going "simplex" (both stations on same frequency in a different part of the band) will leave the repeater free for other stations to use that can't establish simplex communications!

Repeater Etiquette and Reporting Emergencies
The first and most important rule before using a repeater is to LISTEN FIRST. Nothing is more annoying than someone that "keys up" or DOUBLES in the middle of another conversation without first checking to make sure the repeater is free. If the repeater is in use, wait for a pause in the conversation (watch your S meter and wait for it to drop indicating the repeater is listening) and simply say "Emergency, Emergency, Emergency", and wait for one of the other stations to acknowledge your call. If for some reason you are not heard, then repeat the 3 "Emergencies" again...then if you are still not heard, try another nearby repeater.

This is not CB radio!
Don't use CB lingo on any ham band such as 10-4, what's your 20, etc.....don't say BREAKER! 
Using the words BREAK, or BREAK, BREAK or BREAK, BREAK, BREAK or any combination of them on Ham radio can be misunderstood by an operator depending on his experience. 
The word "break" or combinations of it carries many different meanings in the ham community and in the English language.
According to THE EMERGENCY COORDINATOR'S MANUAL Edited by Steven Ewald, WV1X and Published by The American Radio Relay League, Inc.,
Quote from the "General Procedures section....

"16) The word "break" is never used UNLESS there is an emergency."

Then further down in the manual, it appears to contradict or discourage the use of the word/s BREAK in the above statement:

"Note: The practice of using "BREAK" or "BREAK BREAK" to announce distress traffic should be strongly discouraged; it has no universally understood meaning.

So rather than have confusion...use plain language!

Many hams use the wording, "BREAK, BREAK, BREAK", (the word "break" repeated 3 times in a row). This is accepted practice on the hf bands where noise may be a problem but on repeaters, usually noise is not a problem, so using "plain" language such as "EMERGENCY", REPEATED 2 OR MORE TIMES can be used to announce that there is an emergency and the frequency is needed to relay vital information....if you hear an "Emergency" call during your conversation with another station....stop transmitting, listen....and then acknowledge the station calling the emergency and let them have the frequency immediately! Don't delay them by saying something on the order of "Stand by breaker" and then carry on your conversation with your contact. Seconds wasted doing this may COST a life!  Listen to them carefully and write down the details of their emergency. They will give you the details of the emergency. Then pause for a moment and wait before you go back to him.......many other hams who heard the emergency call may be responding ALL at the same time.
If someone "beats" you to getting back to him, let him take over. Do not break into the conversations UNLESS there is a need for a relay. Under certain situations due to distances involved with mobiles and repeaters, you may be able to hear a mobile BETTER than the repeater on the input frequency of the repeater. It is a good idea to monitor the input if possible if the station reporting the emergency is having trouble getting into the repeater. You may be closer to him than the repeater and can hear him better!
Whether or not the station reporting the emergency is a base station OR mobile, try to monitor the input of the repeater if there is difficulty in the emergency transmission.

When using VOICE,  use the international standard "MAYDAY" or universally understood "EMERGENCY" to announce traffic of life-or-death importance.
The procedure should be:
1.Select the repeater frequency.
2. Wait for a space between transmissions if the repeater is busy.
3. Key your mic and state..."Emergency, Emergency, Emergency" unkey.
4. Wait for a response from the repeater users. If you get no response, try another repeater.
When you do make contact, state your call sign and give as many details as to the emergency as possible. Don't panic, speak slowly and clearly so the details will be understood the first time! Always give details as exact and specific. Give the details of the exact LOCATION of the emergency using enough description of the location so it can be found easily by first responders. Don't say....on highway 60 and leave it at that. The emergency vehicles need exact locations if at all possible. Remember, seconds or minutes saved equal lives in many cases!
Give number of "victims" if possible. Is there is fire involved, downed power lines, immediate road blockage due to wreckage creating further dangers? DETAILS, DETAILS, DETAILS.
The person on the other end of your transmission is most likely copying the info to paper so he can relay it to the appropriate authorities. Help him hel