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When we want to use classifiers like SVM, we first should extract descriptors using algorithms like SIFT.

But I have a question which might call a silly one: Let's assume $$ \begin {equation} D_a=\{d_{a,1},d_{a,2},d_{a,3},...,d_{a,M}\} \end {equation} $$ where M is total number of descriptors and $d_{a,?}$ denotes the description vector of mth descriptors. My question is: what the sentence "$d_{a,?}$ denotes the description vector of m-th descriptors" means? Does it mean that we have 1000 interest point in the image that each of them has ,let's say, a 128-dimension SIFT descriptors? Do we have 128000 values?

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I think you have a matrix.
Each Row / Column is a descriptor vector of a point in the image.

Just like having features, let's say M features, and each point has M values corresponding to M features.

So each element in the descriptor vector is a value of specific feature for this point.

And yes, if you have M = 128 Features and 1000 points you'll get a matrix with 128,000 entries.

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