Fortuately there is Phil Plait who can come to the rescue:
"The strength of gravity depends on the distance from the source. The closer you are, the stronger the "pull" you feel. The Moon's gravity acts on the Earth; but the diameter of the Earth is large enough in relation to the distance of the Moon that the side of the Earth nearer the Moon feels the Moon's gravity significantly more strongly than the side of the Earth away from the Moon."
And:
"However, there are tides from the Sun, which are actually about half as strong as those of the Moon (the Sun is much farther away than the Moon, but a LOT bigger). This will continue to affect the system, but at this point you're on your own!"
http://www.badastronomy.com/bad/misc/tides.html

My note: There is significant difference between tidal affect and the graviational affect of a body. Tidal affect is related the gravity gradient- the difference between force of gravity related to the distance you are from the gravity source. If in LEO the strongest tidal affect or gravity gradient will be from Earth. You also have a gravity gradient on the Earth surface.
Anyways, the Sun gravity is stronger at Earth distance than the Earth's gravity and is far stronger than the Moon, but Sun's gravity gradient is less steep than compared to the Moon gravity gradient. We don't "feel" the Sun's gravity because we are in orbit. That is if you want measure gravity by an object's orbital speed- the Earth's orbital speed is far faster (29.79 km/sec) than an any object in a Earth orbit (7.8 km/sec).
Now, if you were to "somehow" to become stationary, the Sun "as far as I understand" would not accelerate you as fast as gravity accelerates you on the Earth surface (weight).
Perhaps someone else can explain this. In that regard the Sun gravity is not as strong as earth's on or near earth.
[My stab at it, is that the Earth gravity is "added on" to the Sun's- this rather quickly gets me into "wild ideas", so I guess I'm probably missing something. (this would mean that both the Sun's and the Earth's gravity is "added on" to the galaxy's gravity and that to rest of universe:)]