They were Trees
As far as every definition of the word goes, they were trees.
From their descriptions Telpirion:
had leaves of dark green that beneath were as shining silver, and from each of his countless flowers a dew of silver light was ever falling, and the earth beneath was dappled with the shadow of his fluttering leaves.
The Silmarillion - Chapter 1: Of the Beginning of Days
bore leaves of a young green like the new-opened beech; their edges were of glittering gold. Flowers swung upon her branches in clusters of yellow flame, formed each to a glowing horn that spilled a golden rain upon the ground; and from the blossom of that tree there came forth warmth and a great light.
Laurelin was furthermore also known as the "White Tree" and "Eldest of Trees" amongst it's various other names.
From their creation, everything also suggests that they were quite simply trees that were grown.
there came forth two slender shoots; and silence was over all the world in that hour, nor was there any other sound save the chanting of Yavanna. Under her song the saplings grew and became fair and tail, and came to flower; and thus there awoke in the world the Two Trees of Valinor.
Before the death of the trees Yavanna and Nienna produced a fruit and a flower from Laurelin and Telperion, respectively.
Yavanna and Nienna to put forth all their powers of growth and healing ... Telperion bore at last upon a leafless bough one great flower of silver, and Laurelin a single fruit of gold.
The Silmarillion - Chapter 11: Of the Sun and the Moon and the Hiding of Valinor
Could they be recreated?
The beauty and the magnificence of the trees was the light that was contained within them. Given everything from Tolkien was governed by the fact that the closer a being was to the moment or source of Creation the nobler and stronger that being (and their offspring) was, the Two Trees held a light from the beginnings of Arda and therefore of utmost beauty within them.
There was Light. There was the Light of Valinor made visible in the Two Trees of Silver and Gold. * These were slain by the Enemy out of malice, and Valinor was darkened, though from them, ere they died utterly, were derived the lights of Sun and Moon. (A marked difference here between these legends and most others is that the Sun is not a divine symbol, but a second-best thing, and the 'light of the Sun' (the world under the sun) become terms for a fallen world, and a dislocated imperfect vision).
The Silmarillion - A letter by J.R.R. Tolkien to Milton Waldman, 1951
In the letter above, as well as a note in the letter (which I will add when I get access) Tolkien discusses how the light of the trees was lost and the world had darkened. The two trees had held the ancient and purest light of Middle-earth that had once shined from the two lamps. That light was stolen from the trees by Ungoliant when she had tried to drink it away. What was left of the trees, the last fruit and flower, had been put into the Sun and the Moon, however these were not close to the magnificence of the original trees, and as Tolkien states were only a “second-best thing”.
The trees had a power to make people "greater" and improve their skill and knowledge, even if they hadn't directly looked at them.
... the Edain of old learned swiftly of the Eldar all such art and knowledge as they could receive, and their sons increased in wisdom and skill, until they far surpassed all others of Mankind, who dwelt still east of the mountains and had not seen the Eldar, nor looked upon the faces that had beheld the Light of Valinor.
Were they Maiar, or something similar?
In the Silmarillion, the Maiar are described as follows:
With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree. These are the Maiar, the people of the Valar, and their servants and helpers.
The Silmarillion - Valaquenta, Of the Maiar
The creation of the trees however came far later, after several thousand Valyrian Years had passed, and the Lamps had been raised and felled. The trees were also not sentient, simply being trees within which light rose and waned with a 7 hour period.
From the quote on their creation above, we see that the Two Trees has been grown from saplings and flowered like ordinary trees, Yavanna had imbued her light in them, but they were otherwise nothing different.
Further evidence lies in Ted Nasmith's artwork. Ted Nasmith can be considered an "official Tolkien illustrator having provided artwork for various projects such as The Illustrated Silmarillion etc. In a sketch of the trees, Ted Nasmith provides the following.
Ungoliant and the Two Trees - Sketches by Ted Nasmith
This sketch of Varda and Manwë in Valinor may also contain Laurelin.