In a non auto-ranging digital multimeter, what do the increasing numbers in the Ohm section represent?
The range is set manually  as opposed to auto-rangig.  The number corresponds to the maximum resistance that the meter can measure.  For example, if you dial 2kΩ, the meter can measure the resistance between 0 and maximum of 2kΩ.  If the resistance is above the maximum, the meter will show you some non-numeric symbol, which is usually "OL".

Why are there multiple ranges?  Why not just use the biggest range all the time?  Smaller ranges have finer resolution.  For example, at the 200kΩ range, the resolution is 100Ω.  If you need to be able to tell a 220Ω resistor from a 270Ω resistor, you would need to switch to a lower range.

You can get familiar with ranges by measuring a resistor with a known value.


If you know roughly the resistance of your nichrome wire, you can select the range based on that.  
Alternatively, you can start with the largest range, which will give you the ballpark number.  The ballpark number can be zero.  If the resolution is, say, 10kΩ, then a 10Ω wire will look like zero.  
Alternatively, you can start with the lowest range and dial up the range until the resistance falls into range.


Manual here.  Probably, that's the right one.