I have gone to many websites seeing the speed we would go to Mars. I have gone to this NASA website. They say the average rocket travels at 25,000 miles per hour and would take 2.5 months to get to Mars.
So why when you read about a possible human mission to Mars in the future does it say that it would take 6-8 months to get there?
Is it not true that the engines are turned off after you have gone a ways in space, considering you do not have a lot of fuel? And following Newton's 1st law of motion "An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force." So friction would not explain the difference in travel times.
So could you please tell me why the speed difference is so different?
This is a good question! The NASA site you mentioned is actually kind of misleading, and didn't take into account the fact that both Mars and Earth are moving as the rocket travels. (It says this if you read the "fine print" on the page.) What they did was assume that Mars and Earth were as close as they can be in their orbits, and that both planets were stationary as the rocket moving at 25,000 mi/hr traveled from Earth to Mars. In this case, if you use distance=velocity x time you can find that it would take about 2.5 months.
But, if you tried to do this in real life, your spaceship would arrive to find that Mars had moved in it's orbit quite a bit, and there would be nothing there to land on! Plus, all objects in the solar system are subject to the Sun's gravity and travel in orbits (ellipses, usually), which means that you don't just travel in a straight line from one place to another. So, the catch is that you have to aim for where Mars is actually going to be at the time you expect to arrive, and travel on a portion of an elliptical orbit, and that takes longer.
Usually when people calculate the time for travel to Mars, they consider a special type of orbit, called a Hohman transer orbit. This orbit is the lowest energy way to get from one planet to another. (Less energy means you don't need as much rocket fuel, which is good.) The time it takes to travel to Mars on this type of orbit is about 8 months, which is where that number comes from. You don't have to travel on this type of orbit - there are lots of possible orbits, but this one requires the least rocket fuel. You can get there faster if you are able to use more fuel and can pick a different orbital path. Also, the exact travel time will change a bit depending on the precise Earth-Mars geometry, and the flight path you take also depends on where on the planet you want to land and what speed you want to be going when you arrive. As far as I can tell, most NASA missions seem to use either a "Type 1 interplanetary trajectory", which is faster (about 7 months) and travels less than 180 degrees around the sun, or a "Type 2 interplanetary trajectory" which travels more than 180 degrees around the sun and takes longer (more like 9 months). You can also look at the orbit diagram for the Mars Odyssey spacecraft.
Anyway, that's why there's a difference between the travel times. I don't know what your background is, but there's a website that uses Kepler's laws to derive how long it takes to get to Mars on a Hohman orbit. Another site discussing orbital trajectories is the Basics of Spaceflight site at the Jet Propulsion Labratory.