The problem with your code is that the if statements should be like this:
if color1 == "Brown" or color1 == "brown":
    ...
This is because "brown" is a non-empty string, so it returns True, which means your if condition will always be true. So, it keeps entering the if statements till it reaches the last one.
Here's how we'd correct this particular mistake:
1. brown1 = 5
2. red1 = 6
3. orange1 = 3 
4. color1 = input("Color on resistor? (separated by comma) : ")
5. if color1 == "Brown" or color1 == "brown":
6.     color1 = brown1
7.     if color1 == "Red" or color1 == "red":
8.         color1 = red1   
9.         if color1 == "Orange" or color1 == "orange":
10.             color1 = orange1
11. print(color1)
Let's try to dry-run this program (manually go line-by-line to see what's happening). if color1 = "brown" then we'll enter the if statement at line 5. Then we'll go to line 6. This means color1 is defined to be 5 now. Thus, color1 can never be "Red" or "red". Additionally, if color1 is not "brown" or "Brown", then it'll go to the next line with the same indentation, or 11, which means it'll never check for red and orange.
What you want to accomplish can be done like this:
if color1 == "brown" or color1 == "Brown":
    color = 5
if color1 == "brown" or color1 == "Brown":
    color1 = 6
if color1 == "brown" or color1 == "Brown":
    color1 = 3
Now, a problem with this is that if we change the value of color1 in the lines 2 or 4, the remaining conditions may be true even though you don't want to check them again (For example: if color==1: color=2; if color==2: color=1 resets color to 1 because the second condition is true once more). This is why elif or else if is recommended, making our new program:
if color1 == "brown" or color1 == "Brown":
    color = 5
elif color1 == "brown" or color1 == "Brown":
    color1 = 6
elif color1 == "brown" or color1 == "Brown":
    color1 = 3