Generally, a summon is there to be a meat shield. You want it to take retalitations and attacks, while your real creatures stay safe and then strike themselves.

To do that, you want high speed, good defense melee fighters. They go in before your real troops, take the brunt of the attack and then your creatures mop up the rest.

What you want to avoid are casters, as illusions don't have any mana (that's a weakness compared to nature; summoned nightmares, water elementals or faerie dragons do have mana available)

In cases, where the enemy army consists of fairly slow melee creatures, you may want to clone a ranged creature to get more firepower.

But otherwise these creatures are your best bets:

• Griffin - fast, durable and unlimited retaliation
• Ghost - deceptively durable due to their defense, their low hp will summon you a lot of them
• Air elemental - see above
• Vampire - can sustain himself, also fairly low hp, so higher numbers. For vampires to work, you need to summon enough to be a threat, though.
• Minotaur - their block special is very good and can help them survive several attacks even if their number is low
• Efreet - extra damage due to the flame shield
• white tiger and nomad - very strong lvl 2 creatures

I'm not a fan of summoning lvl 4 creatures with illusions, that's another weakness compared to nature. You get 0.2 griffins per level, yes, but you also get 0.143 phoenixes, That amounts to 39 hp summoned per level and outclasses phantom image pretty much immediately. With hit point based summons, I think you are better off summoning larger numbers of weaker creatures.

Also keep in mind that summoning spells on heroes usually only make sense in campaigns. On a single map the hero will have trouble keeping up with army growth to summon an amount of troops significant enough to have an impact and espeically with order your are then better off using other control spells like blind or berserk.

It depends on the circumstances

@Dulkan gives a good list of criteria to choose the creatures, but I wanted to check if his list of creatures for every specific listed purpose is actually accurate. For this I have made a Google Doc Spreadsheet containing all of the creatures. It has three very important stats calculated for every creature:

1. Effective Damage, equals Attack*Base Damage (due to how damage formula works in this game), representing creature's ability to attack in a lot more clear way than Base Damage or Attack alone.
2. Effective HP, equals Defence*HP. Represents creature's ability to withstand normal attacks in the same way, but keep in mind, again, that magic bypasses Defence.
3. Effective Damage per HP, represents how much attacking potential do we get for each spell.

If you want, you may print this spreadsheet for quick reference during battles, or use it in any way you want. So, let's start.

Summoned creatures as attackers

For this we have to sort the creatures by Effective Damage per HP. Top creatures are:

1. Behemoth, with its 10.05 effective damage/HP! If you fight Barbarians, just use your Genies to clone dat thing, and it will reck. Statistically it's the strongest creature for cloning.
2. Crusader, 8.8, mainly due to the Two Attacks special and relatively low HP. Note that his defence isn't too high, and his second attack comes after retaliation, so try to use something else to soak it. Don't forget to Bless him if you can!
3. Black Dragon, 8.25. Very sad that we can't really summon him, as he is immune to magic.
4. Evil Sorceress, 7.56. The primary "strength" in this case is very low amount of HP for a level 4 creature. Paired with the Teleport special ability, this makes Evil Sorceress a very good target for cloning, likely slaying even a hidden target like enemy Order Mage. Even though we don't get the spells, stats are good enough to use it.
5. Gargantuan, 7.25, arguably the best shooter in the game, and the best shooter to be cloned, again, mainly because he shoots twice. Also note that he has an AoE attack.
6. Dark Champion, 7.2. Not only is the damage already quite high by itself, but it also amplifies with Charge. Sadly, we don't get the Terror spells, but this creature is good as it is. Regeneration is also a good thing: if you summon 350 HP of Dark Champions, you have 2 fully healed Champions buy the time they act.
7. Sea Monster, 7, plus ability to devour an enemy. You can't get them normally in your army unless a map is set to give you some, but if you are fighting them, clone them.
8. Dragon Golem, 6.95. Negate First Strike and First Strike make for a good tactic when fighting one single stack: don't attack the target, but rather choose Defend action and rely on retaliation. This way our mighty Golem gets less damage. A good thing is that we can find Dragon Golems in out castle (unless building them is disallowed on the map).
9. Angel, 6.85. A statistically good flying creature, without ability to use Resurrection in out case. A steady choice, but Dragon Golems are better, so use them if you can. The best flyer to clone among those that can be found in towns.
10. Mantis, 6.8. Fly, stats and binding ability make it a very powerful choice. Though, if our enemy has Mantis to clone, that might be a huge problem sometimes...
11. Devil, 6.7. Teleportation, paired with good damage, is powerful, and while they deal less damage per HP than Evil Sorceress, they are less fragile with their Defence of 33, compared to Evil Sorceress with 28
12. Titan, 6.4. The best shooter to clone among Order units.
13. Frenzied Gnasher, 6.33. Another would-be powerful choice, if only he wasn't immune to magic.
14. Elf, 6.22. If you are fighting Nature and need a shooter for some reason, that's a good choice to clone. Pair it with Forgetfullness cast on enemy units.
15. Thunderburd, Vampire and Bone Dragon, 6.
    1. Thunderbird: Remember about the good special ability that bypasses Defence and deals plain 30 HP of damage per Thunderbird. A descent choice when fighting someone with big Defence, but a weak one overall.
    2. Vampire: good specials allow the vampires to last way longer than other summons.
    3. Bone Dragon: effectively no retaliation, better Defence than the Vampire, but overall, in my opinion, a weaker choice.
16. Phoenix, 5.6.
17. Hydra, 5.28. Deals way more damage if you surround it with enemies and even more if you make one of them attack the Hydra.
18. Venom Spawn, 5.2, poisoning is a thing to consider.
19. Champion, 5, possibly more due to Charge. Possibly way more than the Angel.
20. Cyclops, 4.73.
21. Efreet. 4,68. Notice how low on the list it is.

All of the Order creatures by Effective damage per HP:

1. Evil Sorceress
2. Dragon Golem
3. Titan
4. Naga
5. Gold Golem
6. Dwarf
7. Halfling
8. Genie
9. Mage

Summoned Creatures as meat shield

In this case you just need the creature with the best Defence. Note that their defence is reduced by 20%, so they receive 25% more damage (1/0.8=1.25), so summons are not always the best meat shields. So, let's look at them:

1. Megadragon, 50. Well, if you find one, try to clone him, because even the not-full HP dragon that you can summon damages as if his HP was full. But be careful with retaliations damaging your own troops.
2. Black Dragon, Dark Champion, Frenzied Gnasher, 40. Dark Champion is just perfect, the other two are unavailablae due to being immune to magic. Probably even swallow the morale penalty and have a few in your army for cloning purposes -- if you can get them.
3. Behemoth, Sea Monster, Dragon Golem, Angel, Mantis. Pick the one that suits the current situation most. I would choose Mantis as meat shield for Binding purposes
4. Devil, 33.
5. Gargantuan, Titan, Air Elemental, 32.
6. Thunderbird, Vampire, Bone Dragon, Ghost, 30. The Vampire arguably is the best because of the ranged capabilities and ability to suck life.
7. Faerie Dragon, 29. Overall a very weak choice, as his attack abilities rely on spells.
8. Evil Sorceress, Phoenix, Catapult, 28.
9. Hydra, 26. Possibly better than many others due to also having powerful retaliation. Venom Spawn and Goblin Knight have the same stats, but aren't as good at attacking.
10. Champion, 25.
11. Crusader and Eefreet, 23. Remember, again, that Crusader also has second best effective damage per HP.

I think, time to stop now.

Creatures with some notable specials

These creatures are worth summoning just for special abilities that work in any quantity.

1. Mantis and their Binding
2. Mermaid and their Hypnotize
3. Sea Monster and their Devouring
4. Imps and their Mana Leech
5. Ghosts and their Aging
6. Minotaurs and their Block
7. Evil Eye and their Harmful Spells

What goes to Nature summons compared to Order

On this list in the same file I have made some calculations, and it turns out that while Nature Druid actually summons more creatures than Order Mage of the same level, Genies in quantities big enough can tie with Druids and even overthrow them.