In a standard cryptic crossword grid, there are no "islands". All the words in the grid connect with each other.
Of the two grids below, the one on the left is a connected grid, the kind you would find in a regular cryptic crossword. Pick any two white squares in the grid and you can find a way through the maze to reach one white from the other, without passing over a black.
The grid on the right is not fully connected; the top-right and bottom-left corners are cut-off from the rest of the puzzle. You'll never really encounter unconnected grids of this kind in cryptic crosswords.
Degrees of Connectivity
All connected grids are not equal – some have weaker connectivity than others. If different parts of the grid are linked to each other with just one or two whites, then the connectivity is weak.
A weakly connected grid is a flawed grid.
Why is weak connectivity a flaw?
A grid with weak connectivity can create an imbalance in the way the solutions get filled in. You might have solved large parts of the crossword fully, and yet have empty patches on the grid with no checking letters for help.
The next grid illustrates this. The corners of the grid are quite independent of each other. Even when 3/4th of the puzzle is done (as shown by the yellows), you can end up with an empty top-right side - in no better situation than if you hadn't started with the puzzle at all.
Compare it to the next grid with better connectivity, in which the word lengths and arrangement ensure that the grid gets filled up evenly.
Connectivity vs Checking
A crossword can be well-checked but poorly connected. The grid above labelled "weak connectivity", for example, follows rules for fair checking but is not well-connected. Similarly, it is possible for a well-connected grid to be poorly checked.
A good crossword grid will have a combination of all – pleasing symmetry, fair checking, and strong connectivity.