What it actually governs
The first house is about arrival rather than identity. It describes the manner you turn up in, the physical body you turn up in, and the reflex that fires before you have decided how to behave. People often confuse it with the Sun, but the Sun is what you are driving at over a lifetime; the first house is what happens in the first four seconds.
Traditionally it also covers the body itself: build, colouring, the way you hold yourself, and the ailments you are prone to. Modern practice keeps the presentation and mostly drops the medical claims, which is sensible.
Why the sign on the cusp is your rising sign
The sign occupying the first house cusp is what everyone means by a rising sign or ascendant. Because the earth turns once a day, that degree moves through the whole zodiac every twenty-four hours, at roughly one degree every four minutes. A birth time recorded to the nearest hour can land you in a different sign entirely.
In whole sign houses the situation is simpler: the sign holding the ascendant becomes the entire first house, so the cusp degree matters less. In Placidus and the other quadrant systems the cusp sits at the exact ascendant degree and the house has an uneven width.
A planet in the first house
Anything sitting here is broadcast. A first house Mars reads as forwardness whether or not the person feels forward; a first house Saturn reads as reserve even in someone perfectly warm. The planet becomes part of the packaging, which is why first house placements are the ones strangers describe accurately and close friends argue about.
Bodies close to the ascendant degree itself are the loudest. Within a few degrees the planet effectively merges with the rising sign, and the two are read together rather than separately.
When it is empty
Most first houses hold nothing at all, which says nothing bad. An empty first house is read through the sign on its cusp and the planet that rules that sign, wherever that planet happens to sit. If Libra rises, Venus is the ruler; wherever Venus is in the chart becomes the address the first house reports to.
The seventh house across from it
Every house has an opposite, and the pair is read as a single axis. The first house faces the seventh, which covers partnership and the other person. What you present, and what you seek in someone else, sit on the same line for a reason: the qualities you leave out of your own presentation tend to be the ones you look for in company.
Questions people ask about this
Is the first house the same as my rising sign?
Nearly. The rising sign is the sign occupying the first house cusp. The house is the space; the sign is the flavour it carries.
Do I need an exact birth time?
For this house, yes. The ascendant moves about one degree every four minutes, so an error of two hours usually moves it a whole sign.
What if my first house has no planets?
That is the ordinary case and carries no penalty. Read the ruler of the sign on the cusp instead, wherever it sits in the chart.