The 1st house is the house of the self, and the sign is called your Rising Sign (since it's the sign on the horizon at the moment you were born from the location you were born at). Your Rising shows how you come across, the way others see you, your external self. The Sun is your internal self, the true you, while the Rising is your face. As we all know, most of us don't come across the way we really are, and astrologically, this is explained by having two different positions.
Some people do have the same Sun and Rising - they are the ones more likely to come across as their true self and put their actual selves out there. But even then, it can get muddled depending on planets in the 1st house.
When a planet is in your 1st house, it's sometimes called "Rising". That planet in the 1st house impacts how you come across. Sometimes the planet in the 1st house even overtakes the Rising sign, and people will think your Rising sign is the sign that 1st house planet rules rather than the actual Rising sign. This can especially be the case when the planet is conjunct (aligned with) the 1st house cusp (or Ascendant).
Planets & the Signs
Let's review which signs the planets rule:
Mercury: Gemini & Virgo
Venus: Taurus & Libra
So, if you have any of these planets in your 1st house, the sign that planet rules influences how you come across. It may play with your Rising sign, or it may overtake it.
Planets Rising in the 1st house:
The Sun: The Sun Rising in the 1st house means you tend to present who you really are externally since the Sun is your true self. This may be easiest when the Sun and Rising are the same sign, and less so if they're different. With the Sun ruling Leo, you may seem more gregarious, fun-loving, creative, and friendly, no matter your Rising sign.
The Moon: The Moon Rising in the 1st house means you may lead with your emotional foot since the Moon rules your emotions. You can be outwardly emotional, and not hide your feelings very much. You may seem nurturing and supportive, sensitive and encouraging. The Moon is a fluid planet, so you may also go with the ebbs and flows of life, or it may feel like things ebb and flow a lot for you.
Mercury: Mercury Rising in the 1st house means you may lead with your mind. You can be more engaging, chatty even, and may seem very intelligent. Mercury rules two signs, Gemini and Virgo, so which may prevail for you depends on the rest of the chart (if it's water/earth heavy, then Virgo; if it's air/fire heavy, then Gemini).
Venus: Venus Rising in the 1st house means you may come across as very pleasant, diplomatic, and charming. A 1st house Venus is also usually quite attractive, even if not in a conventional way. Venus rules two signs, Taurus and Libra, so which steps forward for you depends on the rest of the chart (if it's water/earth heavy, then Taurus; if it's air/fire heavy, then Libra).
Mars: Mars Rising in the 1st house means you may come across as highly active, on the go, fiery even. You may be very assertive and confident in your views and actions. You're likely very competitive, and want to win. You can be enthusiastic and easily excitable, and others may find you to be a little bit much with how much energy you have at times (rambunctious!). The color red also tends to be important for you (in your coloring, or worn).
Jupiter: Jupiter Rising in the 1st house means you may come across as very gregarious, always the optimist, and very easy to get along with. You may be easy going and feel like things will always work out, so why rush or worry? You can seem lucky to others, and your breezy attitude can make it seem like you don't work hard for anything (even if you do).
Saturn: Saturn Rising in the 1st house means you may come across as very disciplined, hard-working, and ambitious. You can lead with your professionalism, and you may always be seen as "the adult in the room" (even when young). Others may look to you to take over in challenging situations. You may seem mature beyond your years when young, and youthful when old.
Uranus: Uranus Rising in the 1st house means you may come across as eccentric, unconventional, and a rebel. You can present your quirks and differences with pride, and embrace innovations and inventions. You may alter your outward appearance continuously, or opt for something that would be deemed odd for your culture/society.
Neptune: Neptune Rising in the 1st house means you may come across as compassionate and helpful. You may seem like someone who is very accommodating, and you can be kind of ethereal. Different people may see different things in you, and may project on you, which can be annoying. You can be seen as a spiritual leader or guru. Neptune also likes to give really pretty or soulful eyes!
Pluto: Pluto Rising in the 1st house means you may come across as strong, forceful, and dominant. Others may be intimidated by you, even if they physically tower over you or professionally outrank you. Your stare can be frightening, yet there can be something magnetic about you. You may be hard to take down, and always come back from defeat with lessons learned.
Suggested Reading: The 1st House page for general interpretations