Whether constant acceleration and zero acceleration are the same

[PeterO]

... If it is how a body can move with constant velocity without acceleration. It should have a positive acceleration to move with constant velocity. Hence 'a' cannot be zero. Therefore, there is something wrong with my understanding...

Yes - there is something wrong.
Acceleration is the rate of change of velocity (speed and direction) If we consider motion in a straight line, the direction part can be defined as positive or negative.
Position, velocity and acceleration are each particular measures of a body. Let's consider morning traffic on a freeway - as seen by an observer (you) on an overhead bridge. Position zero is directly below you - positive direction in in the direction you are looking while negative direction is behind you.
Time zero is when you open your eyes. (you have already walked onto the bridge, faced along the freeway and closed your eyes in preparation).
At time zero (when you open your eyes) you see many cars traveling in the positive direction, and many more traveling in a negative direction. Hopefully, none of them will have zero velocity. However, some of them may have been at position zero the instant you opened your eyes (but didn't stay there)
Many of the cars will have had zero acceleration - all those traveling at constant speed in the direction they are heading.
Some cars may have positive acceleration: includes those traveling in a positive direction (away from you in the direction you are looking) and speeding up, but also those traveling towards you (as you look) and slowing down.
Some cars may have negative acceleration: includes those traveling in a positive direction (away from you in the direction you are looking) and slowing down, but also those traveling towards you (as you look) and speeding up.
Those positive and negative accelerations might be constant but don't have to be.
Every vehicle changing its speed (even slightly) is accelerating. All those traveling at constant speed have zero acceleration. (NOTE: I can use the term speed here because we are only considering motion on a straight section of freeway)

I hope that explains some of this.

 

[PeterO]

In the equation v=u+at, u=5 m/sec and a=0, then v=5 m/sec. That means the body is moving with a constant velocity of 5 m/sec. In a velocity-time graph, the equation produces a straight horizontal line. Some texts say the body is moving with zero acceleration and others say it is moving with constant acceleration. Which is correct?

Can you please check those texts. I would think it is more common for some texts to say the body is moving with constant acceleration, while others say it is moving with constant velocity. (or perhaps constant speed, because they are considering motion in a straight line, and always in the one direction).
Sure zero acceleration is an example of constant acceleration, but it is such a special case that we usually like to emphasis that value of zero.