It's a simple matter of definitions. In either direction, there is a voltage above which the diode begins to conduct a large current for a small increase (or decrease in the reverse case) in voltage. The finer details of the current-voltage function in each direction are somewhat different, but as a first order approximation, above a minimum (reverse breakdown voltage) and below a maximum (forward voltage), a diode does not conduct at all, and at voltages below or above these limits, it conducts a lot. This approximation is sufficient for most engineering purposes.
The reason for the difference in terms is that the underlying physical mechanism is quite different. The forward voltage has to do with the nature of the semiconductor, and for all silicon PN diodes, this will be in the neighborhood of 0.65V. The reverse breakdown voltage additionally depends on the geometry and design of the device, and quite a range of values are attainable, even among silicon PN diodes.
Also, don't let the term "breakdown" suggest that the diode "breaks". What is "breaking down" is the usual state of the diode that prevents reverse current flow. Once the reverse breakdown voltage is exceeded, the diode isn't necessarily damaged. However, a large current will flow, and if this current isn't limited (say, by a series resistor), then the diode will overheat. Then it will be damaged.
Note this isn't really any different from the case when the diode is forward biased. Any attempt to apply significantly more than the forward voltage will result in a very large current which overheats the diode and destroys it. Limiting the current avoids damage.
Ordinary silicon diodes (example, 1n4148) are not often intentionally operated in reverse breakdown. Their behavior in this mode of operation is not usually specified except for some minimum reverse breakdown voltage. There are other diodes, such as Zener diodes, which are usually operated in reverse breakdown (though the physical mechanism is somewhat different). These diodes have more completely specified behavior in this operation, because by virtue of their design, the relevant operational parameters can be more predictable and stable.
There's not just one "ideal" diode.
The simplest type (what I think you are talking about) has 0V across it if the polarity is such that current would flow in the direction of the arrow, and doesn't conduct at all if the polarity is reversed.
A slightly more complex model has a fixed voltage drop (usually taken to be 0.6V or 0.7V) if it is conducting current, and does not conduct current if the voltage is less than the fixed drop. If you consider the voltage drop to have a temperature coefficient (around -2mV/°C) then you can roughly incorporate temperature effects.
In either of the above two cases, to find out what the diode is doing, remove the diode as a thought experiment and see (it may be obvious or you may need to resort to math) whether the bias would be positive or negative.
An even more complex type of ideal diode obeys the nonlinear and temperature-sensitive Shockley diode equation, with ideality factor 1. Most diodes are not that ideal, and the ideality factor is more like 2. You can also have different Is for positive and negative bias.
Finally, you can include ohmic resistance in the equation (which has a positive temperature coefficient), and you'll be fairly close to modelling diode behavior in most real situations.
Best Answer
The books just present a simple model.
It really never is 0.7 V and will vary under many conditions, such as current through it and temperature.
Voltage will be higher at higher currents due to resistance.
Voltage will be lower at lower currents due to exponential voltage vs. current characteristic, where the resistance doesn't play a significant role.