JavaScript on Firefox, 69 bytes

WARNING: This script may hang your browser! Do NOT test it with other browsers (e.g. Chrome) as it may crash your tab / browser.

for(console.log(a=,i=99);--i;a=a.a=k:new Int8Array(1e9));alert(1)

How: When debugger is enabled with console tab activated. console.log makes memory leak possible, and the program will run out of memory soon. When debugger is disabled, GC would clear memory allocated, and you would see an alert with text 1 finally.

T-SQL, 6 bytes

sp_who

Always returns an extra row under the executing user's name when the debugger is active.

MSVC (64-bit), 50 35 bytes

main()return IsDebuggerPresent();

Exits with code 0 normally but 1 under the debugger (note that WinDbg does not automatically display the exit code but you can fake it by examining the stack). Edit: Saved 15 bytes thanks to @Ruslan.

APL (Dyalog Unicode), 0 bytes

In Dyalog APL, debugger vs no debugger is choosen by running one's workspace in either the IDE interpreter or the runtime interpreter.

Loading a completely empty default settings workspace into the IDE interpreter, immediately drops the user into REPL mode as applications must actively shut down the interpreter for it to quit:

However, the runtime interpreter does not have a REPL mode, and so if an application comes to an end (for example because it is completely empty) and does not explicitly shut down the interpreter, it implicitly has attempted to reach REPL mode ("the APL session"), and complains:

Python, 29 bytes

import sys
sys.modules['pdb']

Without debugger (python3 /tmp/foo.py):

Traceback (most recent call last):
 File "/tmp/foo.py", line 2, in <module>
 sys.modules['pdb']
KeyError: 'pdb'

With debugger (python3 -m pdb /tmp/foo.py):

(empty output)

because pdb is loaded into an app-visible module in the second case.

Java, 131 102 bytes

v->(java.lang.management.ManagementFactory.getRuntimeMXBean().getInputArguments()+"").indexOf(":jdwp")

When running without a debugger this function returns -1. When running with a debugger, this prints a positive integer.

Try it online!

This technically detects if you could attach a debugger; not if one is currently connected... Not sure if that's valid

-23 bytes thanks to Olivier Grégoire for mentioning that I can cut down on what I search for in the runtime parameters and for lambda-izing my answer

-5 bytes thanks to Benjamin Urquhart for noting that calling toString is dumb when I can let Java implicitly convert

JavaScript, 17 bytes

debugger
alert(1)

debugger stops execution if the debugger console is open, else it does nothing

Robotalk, 22 bytes

debug l: chronon l ifg

If the debugger is active, the debug instruction pauses execution until the end of the current game tick. In this case, the variable "chronon" is always 1 or greater when read, and the program executes an infinite loop. If the debugger isn't active, debug is a one-cycle no-op. "Chronon" is now 0 when read, the "ifg" branch isn't taken, and execution reaches the end of the code, which is an error condition.

MATLAB, 6 bytes

dbquit

When stopped in the debugger, this will quit debugging mode.

When not under debug, it will print an error message that it cannot be used when not stopped in the debugger.

It should be noted this only works for evaluation in the command window (one of the three ways of running MATLAB code). It won't work in a function or script as once you try to run the script or function, you stop being in debug mode during execution.

The closest you can get to something that will run in a function/script as well would be:

MATLAB, 22 bytes

feature('IsDebugMode')

This is an undocumented command in MATLAB. If you are stopped in the debugger in a script or function, and then call another function (or command evaluation) that contains the above line, it will return true.

Outside the debugger it will return false.

If you were however to just run a script/function containing the above, even if you add a breakpoint at the start of the script/function and step through, it will return false, because again, once you start executing you stop being in the debugger temporarily.