What does "line level" mean?
A device that operates at line level either has a very strong output signal, or only functions properly when you feed a very strong signal into it. Examples of line level outputs include mic preamps, mixers, the "line out" of an amp, and some effects-loop "send" jacks. Inputs needing this level include power amps, most rackmount signal processors, and some effects-loop "returns". This is in contrast to "instrument level" which is what typically comes direct from a guitar or bass, and "mic level" which is the typical output of a microphone or DI box. Both are much lower than line level.
Generally speaking if you send an instrument-level signal into a device that needs line-level input, you will get weak sound, inadequate processing, and probably extra noise as you boost the signal to compensate. If you send a line-level signal into a device that's meant for instrument or mic-level input, you will get distortion. The effects loop on many amps is designed to both send and receive line-level signals, so putting a typical pedal in the loop will often get noise, weakness, and distortion. You may find some exceptions though: either an amp loop that can operate at instrument level, or a pedal that can operate at line level.
The "loudness" or "strength" of an audio signal inside your rig is measured in AC voltage. However the numbers you'll read in an amp's manual or on a website are usually given in dB or dBu, not voltage. The term dB (decibel) by itself means the amount a signal level changes in relation to wherever it started. When you see gear specs that say "-10 dBv" or "+4 dBu", they are telling you how much lower or higher the average output is relative to a specific fixed reference voltage. That voltage is usually either 1.0 V, referred to as "0 dBv", or 0.78 V, referred to as "0 dBu". The terms dBv, dBu, and dBm have different values, but they all have that third letter that signifies a specific reference point; you can use them to calculate voltage levels.
Some common levels you'll see:
+4 dBu is "professional" line level, common in modern pro recording gear, and it is about 1.25 V.
0 dBv is an average line level, typical output from rackmount guitar/bass preamps.
-10 dBv is "consumer" line level, common with older and cheaper recording gear.
-20 dBu is roughly in the neighborhood of a typical instrument's output.
-30 dBu is again in the neighborhood of a typical microphone or DI box's output.
However, instruments and microphones can have a very wide range of output levels in reality, so it is most practical to think of instrument-level and mic-level in/outputs as just "a lot lower than line level", rather than calculating specific dB amounts.
It may even be necessary sometimes to boost one "line level" output by using another gain stage, for example if the first output is specified for -10 dBv and the device you're trying to drive is designed to operate best with a +4 dBu input level. Remember that decibel numbers by themselves are just ratios in reference to a specific starting point, not a fixed value; in other words, 35 dB gain from one device can result in the same actual level as 50 db gain, or 10 dB, or even -20 dB from another device--it all depends on what values each separate engineer started with. 35 dB gain from a boost pedal is a lot, but it may not necessarily get you up to the +4 dBu level needed to drive most power amps, for example. So look for that third letter after the dB to know that you're dealing with a fixed reference point, and therefore a firm value for the highest average voltage output. +4 dBu is the same level all around the world.