First of all, you need to get a secret developer key from the App Garden
Next, since you've stated you're interested in performing a search, look at the API documentation. You will see several "kits" on the left, and "API methods" on the right. Under the photos method you can see flickr.photos.search, which explains the arguments you can pass to the API, what type of response to expect, etc... Great, so now we just need some example code.
I searched Google for "flickr search php example" and came across this tutorial. The code from this page is provided below for your convenience, and I tested locally to confirm it actually works:
<?php
$api_key = 'your api secret key';
$tag = 'flower,bird,peacock';
$perPage = 25;
$url = 'https://api.flickr.com/services/rest/?method=flickr.photos.search';
$url.= '&api_key='.$api_key;
$url.= '&tags='.$tag;
$url.= '&per_page='.$perPage;
$url.= '&format=json';
$url.= '&nojsoncallback=1';
$response = json_decode(file_get_contents($url));
$photo_array = $response->photos->photo;
foreach ($photo_array as $single_photo) {
$farm_id = $single_photo->farm;
$server_id = $single_photo->server;
$photo_id = $single_photo->id;
$secret_id = $single_photo->secret;
$size = 'm';
$title = $single_photo->title;
$photo_url = 'http://farm'.$farm_id.'.staticflickr.com/'.$server_id.'/'.$photo_id.'_'.$secret_id.'_'.$size.'.'.'jpg';
print "<img title='".$title."' src='".$photo_url."' />";
}
Hopefully this helps you get started. Alternatively, you can grab one of the kits mentioned above and use that to see further examples.