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What you are doing here is titration. You know you have a solution of HCl, but you do not know how much HCl is in it. For this you use something that can react with HCl (NaOH) and use an indicator to tell you when the reaction is complete. The reaction is pretty simple:

HCl + NaOH --> H2O + NaCl

You can see here that NaOH and HCl have a 1:1 mol relationship. So, lets find out how many moles of NaOH you used up with concentration = moles/volume

0.10 M NaOH = moles NaOH/ 0.0197 L NaOH solution

Remember that M is in moles/L

moles NaOH = 0.00197 moles

Since you have a 1:1 relationship of NaOH with HCl, the 0.00197 mol applies to HCl as well.

The next question works the same way, but backwards. Try doing it yourself if you understood the first part before reading my answer.

Find out how many moles of HCl you have so you can find out how much moles of NaOH you need for the neutralization.

0.050 M HCl = n HCl / 0.020 L HCl soln

n HCl = 0.001 mol HCl

Remeber the 1:1 relationship, which gives you that n NaOH = 0.001 mol. Now all you need is the volume.

0.1 M NaOH = 0.001 mol NaOH/ Volume soln

V = 0.01 L = 10 mL

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Q: How do you test hydrochloric acid whether it is strong or weak?
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