I am a part time person who usually works 35 hours a week, We are paid bi-weekly. When I work over 40 hours the first week and less than 40 hours the second, I am only paid for the total hours over 80 that I worked during the past 2 weeks. Is this right?

Generally speaking, assuming you are in the private sector, no this is not legal. Each work week stands alone. If your employer will not deal with this, contact the Texas Workforce Commission and file a claim for unpaid overtime.

Does this apply to a non-profit also?

Yes. Non-profit is an accounting method; it does not mean that the employer isn't required to comply with the same laws as all other employers.