I wanted to add something that might be missing from other answers.
Real and Complex Signals
A signal that you can see on an oscilloscope is always a real signal, meaning it can be represented by real numbers. There's no intuitive way to have a complex number transmitted through a medium, unless we take two signals and interpret them as two components of a single complex-valued function. We can label channel 1 as $I$ and channel 2 as $Q$ and we can imagine the results of signal processing operations when we feed them a theoretical complex signal with these two components, but again, this is a matter of interpretation.
A phasor is a representation of a real signal by complex numbers. This is not the same as a complex signal, it is the representation of a real signal which encodes the amplitude and phase relative to a known reference (usually tacitly assumed, like referring it to time $t=0$ when you have $\cos \omega t$). The Fourier transform of a real signal is also complex (although for the Fourier series we can either use the $\sin$ and $\cos$ form or the complex $e^{jk\omega}$ form) but still, the FT of a real signal is only a complex representation of a real signal.
Basic IQ Modulation
With this background understood: there are two places where you'll often see I and Q. The first is in software-defined radio (SDR) or, generally speaking, anything with a digital downconverter (one example being real-time spectrum analyzers). For everything that happens in the processing chain after the signal becomes an IQ signal, we treat that signal as a sequence of complex numbers. The second is in RF modulation/demodulation schemes (like double sideband suppressed carrier, DSB-SC) where two signals are encoded into a single real signal which can be demodulated to recover both the originals.
It sounds like I'm contradicting myself: previously I said that a signal you see on an oscilloscope is always a real signal, but if I probe the output of a DSB-SC modulator, I can see the signal on the scope, and when I demodulate it I get two data streams back! Is this is an example of a complex signal, with two independent signal components? The answer is again: it's a matter of interpretation. Here's the secret to IQ modulation (as opposed to IQ samples in an SDR): the only way to "recover" the original two signals is to be synchronous with the transmitter's local oscillator. If you're not, then trying to demodulate will just produce garbage, and you'll have no way of knowing what information was there before. Therefore, by locking to the transmitters LO, we can interpret the demodulated signal as if it was a complex signal, because it sort of is, as long as you know what the reference is.
You might reasonably think that if a real signal can encode two other signals, then we can imagine that all real signals are actually complex signals, as long as we know how to use them properly. But then consider the case where four signals, or eight, or 128, are encoded in a single real signal. Maybe use amplitude shift keying, or QAM, or time division multiplexing. Point is: just because we can encode information into a real signal does not mean the signal is complex- the only way to make a real signal complex is to interpret it that way.
Answer to Your Question
When you refer to IQ data, I assume you mean in the DSP context, as opposed to the RF modulation/demodulation context. You can, obviously, perform the operations of a modem with a digital down-converter, in which case the reason IQ data is used is because that's the only way to demodulate. (And IQ modulation is used in general for spectral efficiency, etc, see other answers).
But we can consider this in a different scenario: a real signal is converted to IQ samples because, for DSP routines, we can work with signals with amplitudes and phases (relative to the synthesized LO, which might lock to an external reference or might just be free-running), instead of simply "the most recent $N$ samples" in a vector. That is, to get the instantaneous amplitude of the real signal input to the ADC, we take $\sqrt{I^2+Q^2}$, and to get the instantaneous phase (which keeps increasing and wrapping over and over) we take $\arctan(\frac{Q}{I})$. It's another complex representation of a real signal, and it's mathematically convenient to use, especially when the synthesized LO has some significance in the system (like locking to a transmitter's carrier, or serving as the center frequency in a spectrum analyzer so that the center frequency maps to 0 Hz/baseband).
These two interpretations (IQ modulation, used for its useful RF properties, and conversion to IQ data, used for its convenience) are explained more in @Envidia's answer.
