In simple words
An aspect is an angle between two planets. A square means they are about 90° apart.
It is a challenging angle: the two parts of you pull against each other, which builds strength when you work with it.
In a birth chart
Squares feel like tension or recurring challenge, and they are also where much of our drive and achievement comes from. Working with a square, rather than against it, often builds a remarkable strength.
Vesper counts a square when two planets are within 6° of the exact 90° angle in a birth chart, and within a tighter range for day-to-day transits.
As a transit to your Sun
Sun square your Sun. The Sun is at a sharp angle to your natal Sun, a quarter-way point in your solar year. It tests the direction you set.
Moon square your Sun. The Moon, which moves through your feelings, is at a sharp angle to your Sun, the part of you that sets direction. For a few hours, mood and purpose pull against each other.
Mercury square your Sun. Mercury is at a sharp angle to your natal Sun, so your thinking may clash with your goals or with other people's views.
Venus square your Sun. Venus is at a sharp angle to your natal Sun, so desire for comfort or approval can clash with your deeper goals.
Mars square your Sun. Mars is at a sharp angle to your natal Sun, so energy runs high and can turn into frustration if blocked.