Alternator problem? These are my test results...

My battery was at 13.10V before doing anything. When I started the engine, it quickly went up to14.50V, but quickly went back down to 12.35V a minute later. The battery was not charging, I guess. I let the engine cool off for one minute, and the battery went to 12.51V.

The problem started after I forgot the lights on twice in a month. I was able to jump start the car with someone else's booster twice. But it failed the 3rd time. A mechanic recharged the battery for me.

Did you have any electrical loads in the car turned on when testing? Also have you had the battery tested to be sure it is good?

Your battery may indeed be weak, but I still believe the system voltage should be higher than what you're seeing with a functioning alternator.

The next time you can get it to not charge, try raising the engine speed off idle and see if the system voltage rises above 14V. If it does, that would suggest that the alternator is working, and that it simply can't keep up with a weak battery at idle. If it still doesn't improve, then we can go after a failing alternator.

I'll try to find someone who could keep an eye on the voltage while I raise the engine speed. And I"ll be driving around with it tomorrow. I'll check the voltage after.

Is it a Honda? some Honda USA market Hondas have an odd ball method of controlling the charging voltage to reduce emissions and fuel consumption On Hondas the engine computer can in effect switch off the alternator completely if the charging current drops suggestings the battery is fully charged.
I am 100 % with Tyler so I would also suggest you get the battery tested with a proffesional battery tester as the non-starting symptoms are a good fit for a battery with a dead cell.

12.5 to 12.7 voltls is the normal battery voltage engine off but a battery with a dead cell will usually show normal voltage, but when the battery is put under heavy curent load the battery voltage will drop like a stone.
There are two types of tester used to test for a battery with dead cell the old style puts a huge load on the battery, the more modern type uses high-tech electronic testing.

A quick test for battery health is to measure the battery voltage with engine off and key in off position (everything off). You should have about the voltage Andy mentioned (12.5V~12.7V). Turn on your headlights (hi beams), this puts a decent load on the battery. You should see the voltage drop immediately to 12.1V~12.2V or so and then slowly drop, if it keeps dropping rapidly then chances are you have a bad battery. This test is not as good as a 'real' battery tester but can typically point you in the right direction.

It is not uncommon for batteries these days to fail suddenly. The last 3-4 batteries I have had to replace have seemed to be good one day and died the next day (after 4-5 years of use).

To answer to one person, this is a 2003 Mitsubishi Lancer model.

I will do the test suggested above: With the engine off, I will let the lights on for a while and monitor the voltage. I could not do it today because of the bad weather.

The first test that I did was to remove the positive cable to the battery, and the engine stopped immediately. I've learned after the fact that I could have damaged the electronics. Nothing of that sort happened. But it probably means that the problem is with something else than the battery.

Before doing anything, the battery was at 12:52V. WIth A/C and lights on, and engine on, it stabilized at 13:17V. Stop engined, and turned A/C and lights off, it stayed at 12.44V.

First of all testing the alternator --
Did you test the charging voltage at normal idle ? if so the result is OK but normally it is a fast idle test and the voltage is normally a bit higher .
At idle 13.17 volts with the AC and head lights on the alternator is meeting all the electrical loads and doing some charging of the battery. However this test would normally be done at fairly fast idle 1500 to 3000 rpm. The procedure is normally to raise the idle speed to 2500 rpm or so for a gasoline engine ( diesel 1,700 rpm) with no electrical loads on then switch the electrical loads on then switch them off.
At no point with the engine running should the voltage go outside the 13.1 to 14.9 volts range. Typically expect 13.4v to 14.45v with no spikes over 15v when you switch the electrical loads off.

Now the battery --
All lead-acid batteries like humans decay with age but how fast the decay depends on how they are treated in service, every charge and discharge cycle reduces how much electricity the battery can store when re-charged. If a battery is discharged below 50% and then recharged a measureable ammount of damage is done to the battery's amp hour capacity. Automotive starter batteries don't like this type of treament and a few deep discharges do a lot of damage.

Even normal service the amp hours capacity of the battery declines with age. A typical 2 litre car might be fitted with 65 amp hour battery when new but after 4 years normal service the amp hour capacity might be down to only be 35 amp hours, all will seem normal the car will start fine the voltages with engine off or running will be unchanged no problem will appear until the battery's full amp hours capacity is needed for example if the vehicle left unused while the owner is on vaction. Eventually of course as the years pass the battery will decline more and refuse to hold any charge.

The other main cause of battery death is a dead shorted cell. A battery with a dead cell will at first seem perfectly normal but a random intervals refuse to even attempt to crank the engine. After a boost start if the engine is run for even a few minutes then switched off the engine run and start normally for the rest of the day, week or even month. As with every battery fault the no cranks no starts will become more frequent over time.

None of the three commonly tests used on car batteries are a 100% reliable as indicators that a battery is in good health but a fail on any of the three indicates the battery is dead or dying. Personally I choose to use a high load tester and the more modern electronic AC tester impeadance. The high-tech AC impeadance test gives a fairy good indication the percentage of life left in a battery, and the high current load test is a clear good/bad indication of a dead shorted cell. Most battery sellers will be equiped to do at least one of these tests for you.
A final word on batteries, quality counts if you are planning keep a car for more than two years it pays buy a quality premium grade battery.