When we perceive (or imagine) an object, it has a whole collection of properties. For example, when I look at my car I perceive the following properties in various ways: the color (red), extension (i.e., volume/occupation of space), temporal location, spacial location, shape, texture, temperature, motion, quantity, and sound.
Some of these qualities are subjective in nature and some are objective. That is to say, some of the qualities appear the way the do as a consequence of how my brain is wired and how my perceptual system works, and some are actual qualities of the objects "out there" that I'm perceiving.
If my brain had been wired differently, the subjective qualities might have appeared differently. For example, maybe my car has been sitting in the sun all day so when I touch it, it has the quality of "warmness." But "warmness" isn't an actual quality of my car; that fact that I feel warmness is a consequence of how my fairly standard human perceptual system is calibrated to perceive an object's average kinetic energy (i.e., motion).
We can bring the subjective nature of temperature out by considering what would happen if some aliens from Mercury were to touch my car. Their perceptual system is (most likely) calibrated differently from mine and so, upon touching my car, they might perceive my car's temperature as cold. The central idea is that the warmth or coolness of my car isn't "in" the car--it's in how I perceive the car when I touch it.
Consider again those same aliens perceiving my car. Regardless of how their perceptual apparati are wired, they will perceive my car as existing in a spacio-temporal location, of having a shape, and of being solid. No amount of different wiring could change that. They could fail to perceive these qualities, like a blind person does, but barring blindness, they could not perceive them in any way other than how they actually are in the object.
So, what inference can we draw from this? It seems that there are certain properties that are mind-dependent (i.e., subjective) and other properties which are objective--that is, they are properties of the object itself.
Lets look at one more example of a subjective property: color. Is my car, i.e., the car itself actually red or is the redness a consequence of how my brain interprets certain neuro-chemical signals? While there is a small minority who disagree, most philosophers and perceptual psychologists will say that the object itself isn't red, it only appears red because of the way my brain interprets the electro-chemical signals which are cause by light waves of a certain frequency.
For example, red light has a frequency of 4×1014 Hz. The light itself isn't red, my brain simply interprets light at that frequency as red. However, it's totally conceivable that our brains could have interpreted that very same frequency as purple. The point is that it is our minds (via our perceptual systems) that add color to whatever we are perceiving. Color isn't "out there" it's "in here" (I'm pointing at my head).
So, why does all this matter? For modern philosophers there is a (fairly) clear distinction between primary and secondary qualities. Primary qualities like extension, shape, quantity, and motion exist in the objects we are perceiving; they are part of the physical reality--i.e., they are objective. Any two types of beings who perceive the same object will agree on the nature of that objects primary qualities. For example, it's not possible for 2 people to look at a circle and for one to perceive it as a triangle. Shape is a primary quality; it's not in the mind, it's inherent in the object.
Secondary qualities are color, sound, texture, and smell. These are qualities our mind adds to the objects we are perceiving. Secondary qualities are mental; they do not exist in the external objects of perception.
Anyhow, this should help with understanding some of what Berkeley and Russell are discussing.