AC power adapter

Disconnect the AC power.
Remove the battery.
Press the ON/OFF button to "drain" any residual power.
Open the cover over the RAM slots.
Remove the sticks of RAM, and then re-insert them.
Reconnect the battery.
Press the ON/OFF button.
Does it still beep?
If so, then the CPU or graphics adapter or motherboard is the source of the problem, and you're stuck -- none of these parts are replacement without a certified computer-technician.

BATTERY IS BAD COULD BE CAUSED BY LEAVING AC ADAPTER PLUGGED IN ALL THE TIME ORDER NEW BATTERY COST LOW ON EBAY AROUND 20 BUCKS

Try another mains power cord. If this does not work then the PA-21adapter is faulty and needs to be replaced.

Is the AC Power adapter the original adapter of the laptop? If it is, try to drain the battery and charge it for 8-12 hours without using or opening the laptop just charge it. After charging, remove the power adapter as you open your laptop. Make this way a habbit of using and charging your laptop to make the charging and battery life last long.

if this just happened today try to do a power drain by turning off the computer, removing the battery, removing the ac adapter and then without any power on the computer press the power button for 15seconds without any power on it then after 15 seconds try to turn it on with the use of the ac adapter alone.
- if it turns on let the computer boot up and then shutdown the computer to reconnect the battery and try again
- if it does not work and still the same try to connect the computer to an external monitor
- either it works or not last thing is back up the files because you might need to send the computer in for repair

Dell systems will not charge the battery without a Dell Branded AC adapter connected to the system (The BIOS looks for the adapter ID and disables the battery charger if not ID'd)
It appears, that since the system will not power on with either AC adapter, that there is a problem with the power connection of the system. You can replace the power board
http://cgi.ebay.com/DeLL-Inspiron-1525-1526-Power-Jack-Board-48-4W006-011_W0QQitemZ380212346664QQcmdZViewItemQQptZMotherboards?hash=item58866bff28#ht_953wt_1165
if that fails to resolve next is the motherboard
http://shop.ebay.com/i.html?_nkw=dell+inspiron+1525+motherboard&_sacat=0&_dmpt=Laptop_Adapters_Chargers&_odkw=dell+inspiron+1525+charger+board&_osacat=0&_trksid=p3286.c0.m270.l1311

remove all external peripherals. remove laptop battery. remove ac adapter.
plug ac adapter to wall power socket without the laptop connected check if the ac adapter led will stays on. if yes then connect it to the laptop and check if the ac adapter led turns off. if yes motherboard laptop has problems and need to be replaced.
if ac adapter led stays off without connecting laptop, try different wall power socket, if still ac adapter led is off then replace the ac adapter.
if ac adapter has light stays on then connected to laptop but laptop cannot even power up even the ac adapter led is on then laptop motherboard needs to be replace.

I got one. Thought my problem was the adapter and it turned out to be a picture issue.

Sounds like you have a loose DC jack or a faulty AC adapter. Try testing the adapter with a multimeter (or have a someone do it for you) to see if it's output is consistent. If you wiggle the ac adapter plug slightly when it is plugged into the jack on the notebook and this causes the power to go on-and-off (you should be able to tell by a power/charge led on the notebook) it is likely the DC jack if the AC adapter tests ok.

Prices to replace the jack vary. My shop does it for $65 labor + $15 for the jack on most non-apple notebooks.