There are several ways to consider the houses in Astrology which I've seen:
In one case, there's the general sense, as I'd label it. It is basically whole-house (I won't get into that later-on, but each house is one sign), and is used for general astrology along the lines of the over-all current energy.
In another case, a lot of daily horoscopes which mention houses, even ones which can generate a somewhat accurate birth chart on their site, will start the houses with your sun sign -- which is basically incorrect according to most experienced astrologers. Again, it uses the whole-sign houses, and their site's generated birth chart would even correctly label the rising sign, which is typically the determinant of a person's first house.
As for personal astrology, your own houses are determined usually by your rising sign -- what sign was at the eastern horizon at the time of your birth. This isn't every astrology tradition, though it is widely held in western astrology. I believe Vedic and Draconic astrologies use different determinations, and I won't get into thirteen-sign astrology, which is a whole other can of worms.
Any way, your rising sign determines your first house. Some people prefer different systems such as whole-sign houses, Placidus, and I forget the other main aspect some may or may not prefer. And there's a lot of discussion about why people prefer one over another.
If you're curious about your own birth chart, I'll suggest astro.com though there are others which generate an accurate chart. You'll need your birth time, and as close to the right location as possible. Astro.com offers the birth chart for free, and I believe they also give a brief summary of what is seen as for major aspects. But it's also useful, if you're curious, to search your own aspects such as " [planet] in [Nth] house," " [sign] in [Nth] house" etc.
A quick tip is that the sign at the beginning of the house is the one which rules that house, so if the house intersects Taurus and ends in Gemini, the house is ruled by Taurus -- and the ruling planet of Tarus, though it may be in a different part of your chart, has some bearing on that aspect as well, though I am still a long way from fluency in astrology at this point.
The "first sign" of astrology is Aries. So, when someone is speaking about "the" first house in the very general sense, they mean Aries as the first sign, and on through the rest from there.
Additionally, when consulting a general horoscope, it's often recommended to read both your sun and rising signs. They will both be useful, somewhat like they'll help clarify each other. However, some will insist that the horoscope for the rising sign would be at least somewhat closer to accurate.
