Needleman- Wunsch explained in detail

Global alignment algorithm:

The alignment takes place in a two-dimensional matrix in which each cell corresponds to a pairing of one letter from each sequence.The alignment starts at the upper left and follows a mostly diagonal path down and to the right. When two letters are aligned, the path follows a diagonal trajectory. There are several places in which the letters from TCGCA are paired to gap characters. In this case, the graph is followed horizontally. Although not shown here, the path may be also be followed vertically when the letters from TCCA are paired with gap characters. Gap characters can never be paired to each other.

 Initialization:
In the initialization phase, you assign values for the first row and column . The next stage of the algorithm depends on this. The score of each cell is set to the gap score multiplied by the distance from the origin. Gaps may be present at the beginning of either sequence, and their cost is the same as anywhere else. The arrows all point back to the origin, which ensures that alignments go all the way back to the origin (a requirement for global alignment). If the gap penalty is -2 then it'll be -2, -4,-6...

 Fill :
In the fill phase (also called induction), the entire matrix is filled with scores and pointers using a simple operation that requires the scores from the diagonal, vertical, and horizontal neighbouring cells. You will compute three scores: a match score, a vertical gap score, and a horizontal gap score. The match score is the sum of the diagonal cell score and the score for a match (+1 or -1). The horizontal gap score is the sum of the cell to the left and the gap score (-1), and the vertical gap score is computed analogously. Once you've computed these scores, assign the maximum value to the cell and point the arrow in the direction of the maximum score. Continue this operation until the entire matrix is filled, and each cell contains the score and pointer to the best possible alignment at that point.
The match score is the sum of the preceding diagonal cell (score = 0) and the score for aligning C to P (-1). The total match score is -1. The horizontal gap score is the sum of the score to the left (-1) and the gap score (-1). The horizontal gap score is therefore -2. The same is true for the vertical gap score. Your maximum score is therefore the diagonal score (-1), and the pointer is set to the diagonal

 Trace back :

The trace-back lets you recover the alignment from the matrix. Like the other parts of this algorithm, it's pretty simple. Start at the bottom-right corner and follow the arrows until you get to the beginning. To produce the alignment, at each cell, write out the corresponding letters or a hyphen for the gap symbol. Since you're following it from the end to the start, the alignment will be backward, and you just reverse it. The final alignment looks like this: