Just to expand o nwhat the other posters said...
Der zweite Ding:
What's the point of the republican party?
Like, what is it for? On what base was it founded?
The Republican party was originally founded right before the Civil War. Back then, its main issue was that it was mainly an anti-slavery party (ranging from abolitionists to 'moderates' who wanted to stop the US from having more slave states).
I get impressions from american politics that especially the republican party needs to "rebrand itself" to get more votes
But that's rediculous. The point of a party isn't to have an arbitrary group to achieve power, it's to further a political agenda. Whatever "republicanism" is, the thing the party should aspire to isn't to change it to suit the voters, but to convince voters to follow it
Toughts, authorotative answers?
The party isn't about "republicanism" per se. It's just a generic nice sounding name (inspired, I believe, by the Democrat-Republican party of the pre-Civil War era). It is, as others have said, a coalition of different factions and interest groups in American politics. You can see many of these groups on display in the Republican primaries of 2012 and 2016. You have the social conservative/evangelical wing, with guys like Santorum. You have the hawkish Neocons, like Graham (and Bush). Then you have Rubio who came from the Cuban-American community which has historically been strongly Republican (though that is changing with the younger generation). And you also have guys like Romney and Pataki who represent the more moderate, remnants/spiritual successors of the old group of liberal, Rockefeller Republicans who mainly hailed from the northeast. There's plenty more groups, and of course it's possible for a Republican politician to appeal to different groups at once - Huckabee for instance appealed to evangelicals and blue-collar conservatives.
This is also why sometimes things get weird in local elections, for lack of better wording. That is, for example, you might see some pockets of the Rockefeller Republicans in local politics of the Northeast in places like Maine or Massachusetts, even though they're pretty much extinct on the national level. Or, conversely, you still have the conservative southern Democrats here and there whose positions may or may not be more aligned with the GOP but are Democrats nevertheless (these include guys like the late Senator Robert Byrd, President Carter, and the more centrist Bill Clinton and Al Gore).
The reason why people talk about re-branding the GOP is because it is seen as the party of old white dudes. There is good evidence backing this up, showing that the GOP doesn't do as well as the Democrats when it comes to racial minorities (black, Hispanic, Asian, whatever), women, and the young. There of course are some exceptions - Cuban and Vietnamese Americans, unlike many of their Hispanic and Asian brethren, have tended to be strongly Republican due to their strong anti-communist stances, although this is eroding with younger generations (case in point: from personal experience as a Vietnamese-American, if I were to give really, really rough estimates, I'd say people older than my parents lean Republican like 90% of the time; people around my parents' generation and a bit younger are 60/40 in favor of Republicans; and people my generation are 70/30 in favor of Democrats).
As such, the idea is that the GOP needs to appeal more to these groups if it wants to be able to compete well with the Democrats in the future. For instance, in the 2012 Presidential election, around 75% of Hispanic voters supported Obama over Romney, and the only group that voted more than that were Asians who were a few more percentage points in favor of Obama. Hispanics are an increasingly large voting bloc in many parts of the country, which is why people talk about them a lot. However the other groups matter too - Asians are the fastest growing minority, for instance, and women, well, women are about half the population out there. Of course, one must remember that these are hardly monolithic voting blocs. Take Asians for instance - sure, in 2012 they voted way more in favor of Obama than Romney, but this varied greatly within different Asian groups. South Asians (i.e. Indians, Pakistanis, etc.) voted for Obama like 9 to 1 or 8 to 1. Vietnamese meanwhile were only 44% in favor of Obama. Voters also care about different things, coming back to the issue about different factions mentioned earlier - some Hispanics care more about the economic well-being of the country or social welfare more than immigration (even though immigration is an important topic for them); others worry more about social welfare programs and health reform. And so on.