555 as a Schmitt Trigger and an ON delay timer?
The description was a little ambiguous; this is how I interpreted it:



After the input reaches a certain threshold level (dotted line), the output goes high for 25 seconds.

The functions of a Schmitt trigger and a monostable need to be handled by two separate 555s.  Luckily, you can get two 555s inside one 14-pin IC called a 556.  Here is a circuit which uses the first half of the 556 as a Schmitt trigger, and the second half of the 556 as a 25-second timer.  If you want a different value for the timer, change R3.

Note because of the typical tolerance of electrolytic capacitors, ±20%, the accuracy of the timer will be about the same assuming you use a 1% resistor for R3.  If you want tighter timing, you can get a 10% tantalum capacitor, but they are relatively expensive ($5).  They do make 5% tantalum capacitors,  but only in surface mount packages.



When the input voltage is above 2/3 Vcc, the output of the Schmitt trigger goes low, triggering the monostable. When the input is less than 1/3 Vcc, the output of the Schmitt trigger goes high.  Note that the TLC556, linked to earlier, has a Vcc range from 2v to 15v which gives a lot of flexibility in setting the hysteresis levels.

There are variations on this circuit, including adding a capacitor to the input and a voltage divider to change the trigger voltage.  This article has lots more information.