From the perspective of neuroscience, a "thought" could be described as a structured representation of a point of view, belief, hypothetical possibility, or intended action. A thought is the type of thing that you could write down as a sentence, for example.
We don't fully understand, yet, how thoughts are represented in the brain, but there must be some type of representation. The clues we have so far suggest that some neurons will represent elements of the thought and others will represent the relationships among the elements. The representation is probably highly dynamic, with millions of neurons firing at different times, and the pattern probably never repeats and always advances.
Working memory likely plays a role in capturing different components of thoughts, so that associative areas of the brain can bind the components together.
But this is all fairly wild speculation based on the clues we do have so far.
Consciousness is the "theater of perceptual awareness." It's not clear if consciousness is required for thoughts or not, but it certainly helps. Can you have a thought that you are not aware of? Most people would say it's not a thought in that case. If you're "lost in thought" and someone asks you what you're thinking and your mind "suddenly goes blank", does that mean you weren't thinking anything? If you wake up in the morning with an insight, does this mean you were thinking while you were sleeping?
These questions ultimately come down to what we mean by "thought" in different circumstances and contexts. We will eventually be able to figure out what is going on in the brain in these different scenarios and why; we will then need to decide which types of activity we want to call "thought" and which ones not.
Can the "same thought" be made by many minds?
It's pretty safe to assume that every mind is unique, and that each mind represents "similar thoughts" in different ways. One's own mind may also represent the same thought in different ways as one's thinking evolves. What you "understand" to be multiplication or a calculus integral is surely different now then when you first learned it. The reason is that the building blocks of thought are the representational schemes of the brain - what each neuron represents - and these are changing all the time as the brain learns.