Climbing plants do it in three different ways - self adhering (or self clinging), twining, or scrambling (sometimes called tendril climbers). Those which are self adhering use small rootlike growths to penetrate into a suitable surface, as with Hedera varieties, and some penetrate a surface more than others. Virginia creeper, for example, only lightly adheres to a surface rather than penetrating as deep as something like Hedera (English ivy).
Twining plants, as the name suggests, twine round supports, themselves, or any nearby suitable object, like a bamboo cane or another plant; examples of that type of plant are Honeysuckle and Campsis radicans. Scrambling plants use modified petioles or tendrils to cling onto a support - clematis and sweet pea are good examples of this. Both need thin supports to cling to, such as clematis mesh, thin canes, their own stems or another plant, but the tendrils won't be able to wrap themselves round thick supports, such as those found in trellis, so mechanical intervention by us is needed to persuade them to climb up those, by tying them into place, or wrapping the stems around the wood slats. This method of persuasion is often used for larger, species varieties of clematis which we know will need strong support, such as Clematis montana, which over time develops thick, woody stems.
The term 'climbing' is often used for plants which do not climb at all; it's applied to some roses, pelargoniums and even fuchsia, but none will 'climb' or be high up without being supported or tied to a support. Left to their own devices, they would grow to a certain height and then sprawl, or grow across the ground. The definition of a true climber would be a plant that attempts to haul itself upwards by attaching itself in some way to whatever's available.