Under the influence of gravitational force, the Earth has been orbiting the sun for a few billion years. The moon has been orbiting the Earth for almost as long. As they orbit, every now and then the sun, moon and Earth all line up. The positioning of the moon exactly between the sun and the Earth results in a solar eclipse. And when the Earth is precisely between the sun and the moon, it's a lunar eclipse. Although eclipses look dramatic, they have no influence on gravitational force. The only difference in gravitational force during a solar eclipse is that the moon and sun are both pulling on the Earth from the same side -- but that really makes no difference in any measurable way.
Every object in the universe attracts every other object in the universe. That was Isaac Newton's discovery with the law of universal gravitation. It's a mathematical statement of the magnitude of gravitational force. Newton's equation for universal gravitation states that the force of gravitational attraction between two objects is equal to a gravitational constant times the mass of the first object times the mass of the second, all divided by the square of the distance between them.
Earth, Sun and Moon
The average distance between the Earth and sun is 150 trillion meters, or 1.5 x 10^11 meters. The mass of the sun is 1.99 x 10^30 kilograms, while the Earth weighs in at 6.0 x 10^24 kilograms. The gravitational constant is 6.67 x 10^-11 meter^3 / (kilogram - second^2). So the Earth and sun pull on each other with a force equal to 3.52 x 10^22 newtons. The newton is a unit of force equal to a kilogram-meter/second^2. One newton is equal to 0.22 of the rarely used English unit called pound-force, so 3.52 x 10^22 newtons is 7.9 x 10^21 pound-force.
The average distance between the Earth and moon is 380 million meters and the mass of the moon is 7.35 x 10^22 kilograms, so the force between the moon and Earth is 2.03 x 10^20 newtons (4.5 x 10^19 pound-force). That is, the gravitational force between the Earth and moon is about half a percent of the force between the Earth and sun.
During a solar eclipse the pull of the moon and sun line up so that the Earth feels a combined force of 3.54 x 10^22 newtons (7.96 x 10^21 pound-force) in the direction of the sun. During a lunar eclipse the moon pulls in the opposite direction as the sun, creating a net force of 3.50 x 10^22 newtons (7.87 x 10^21 pound-force) in the direction of the sun.
To put this in perspective, during a year the elliptical shape of the Earth's orbit brings it closer and farther from the sun. When sun and Earth are closest, the gravitational attraction between them is 3.67 x 10^22 newtons (8.25 x 10^21 pound-force), and when they're furthest the attraction is 3.43 x 10^22 newtons (7.71 x 10^21 pound-force). That is, the normal annual variation in gravitational force during the course of a year is more than 10 times as much as the change due to the position of the moon during eclipses.
Gravity on You
Maybe an even more interesting question involves the effect of the gravitational force on yourself during a solar eclipse. The pull of the sun on you is about 0.0603 percent of the pull of the Earth on you. The pull of the moon is about 0.0003 percent of the Earth's gravitational pull. So if you weigh 68 kilograms (150 pounds), at noon during a solar eclipse -- or during any new moon -- you'd weigh 0.6 grams (two-hundredths of an ounce) less than you would at noon when it's a full moon.