What are: Lamrim, Dzogchen, Mahamudra

Lam Rim in Tibetan literally means stages of the path. It's a kind of literature that gives a general overview of the path to enlightenment in terms of the three capacities of individuals and the different practices one must engage in. The first Lam Rim text was Atisha's work called the Lamp for the Path to Enlightenment, and over time other Lam Rim works were written, most famously Lama Tsongkhapa's Great Treatise on the Stages of the Path and Gampopa's Jewel Ornament of Liberation. Both of these list out different kinds of teachings and different meditative practices. They are in many ways quite like the Visuddhimagga.

Mahamudra is a meditative tradition that is most prominent in the Kagyu school of Tibetan Buddhism but is also found in the Gelugpa school, and some forms of it are also in the Sakya school as well. In terms of practices there is Sutra level Mahamudra that isn't secret, and Tantric Mahamudra that is secret. There is also essence level Mahamudra but I have no idea what that is. Mahamudra involved different stabilizing practices to gain concentration and involves meditating on the mind as an object.

Dzogchen is a tradition found in the Nyingma school and sometimes is found in the Kagyu school as well. It has some similarities to Mahamudra but Dzogchen is entirely a tantric level practice and is more secret.

Lamrim is a genre. It is a genre of spiritual literature that gives a thorough overview of the whole teaching, methodically progressing from somewhat crude and primitive concepts of basic levels, to more sophisticated and nuanced perspectives of the higher levels. Each school usually has their favorite lamrim text or two (one for beginning students, one for advanced students and teachers), but if you read them side by side they are all pretty similar in content and sequence, the biggest difference being that of depth and tone. It is not uncommon for a lamrim text to take form of a condensed poem for easy memorization, which then requires a commentary text to hydrate the meaning back to a comprehensible shape.

In Theravada, the idea of stages of the path is not very developed. I'm not talking about levels of mastery of one technique like the 16 stages of Vipassana (there are such texts in Tibetan Buddhism as well, but they are not lamrim) but rather the epic levels that almost completely supersede each other's frameworks, such that the concepts very important at one level look rather naive at subsequent levels.

It is hard to call Visuddhimagga a lamrim text, for two reasons. First, because its scope is limited to what in a lamrim text would be about one sixth of the full path. Second, because it delves very deep into topics that in Tibetan schools usually belong in specialized meditation texts.

The lamrim progression is roughly:

1. "folk Buddhism", with emphasis on faith and basic morality, six worlds, karma
2. the kind of stuff that's studied in Theravada: 4 noble truths, samatha, vipassana etc.
3. mahayana stuff: bodhisattvas, paramitas
4. hardcore madhyamika philosophy
5. the (lower) tantra levels. Yidam generation and energy yoga.
6. mahamudra and dzogchen
7. dzogchen is sometimes separated into its own level above mahamudra

Mahamudra is an advanced-level set of teachings in New Translation schools (Kagyu, Sakya and Gelug). In simple words we could say Mahamudra is the level where we finally get direct experience of what Buddhism is actually about, with no beating around the bush like on the previous levels. That said, for but the most talented students Mahamudra is incomprehensible without the foundation of the previous levels, plus practicing it requires a kind of deep emotional stability that can only be developed through the previous levels.

Dzogchen is a very advanced-level set of teachings in the Old Translation school (Nyingma). In some sense it is very similar to Mahamudra, but it tends to be even more abstract. This is because Dzogchen speaks entirely from the perspective of enlightened Buddha, unlike Mahamudra which still has residual flavor of the student attitude.

If you need a parallel with Theravada, I suppose we could roughly put Mahamudra at the level of Fourth Jhana and Dzogchen beyond Fourth Jhana. I guess I could also say that the practice of Mahamudra and especially Dzogchen is not that dissimilar from Zen although the style is obviously rather different.